Math Sux, Author at Math Lessons

What is a Geometric Sequence?

Geometric Sequence Formula:

a_n=a₁r^(n-1)

a₁ = First Term

r=Common Ratio (Number Multiplied/Divided by each successive term in sequence)

n= Term Number in Sequence

Hi everyone and welcome to Mathsux! In this post, we are going to answer the question, what is a geometric sequence (otherwise known as a geometric progression)? We will accomplish this by learning how to identify a geometric sequence, then we will break down the geometric sequence formula a_n=a₁r^(n-1), and solve two different types of examples. As always if you want more questions, check out the video below and the practice problems at the end of this post. Happy calculating! 🙂

What are Geometric Sequences?

Geometric sequences are a sequence of numbers that form a pattern when the same number is either multiplied or divided to each subsequent term. Take a look at the example of a geometric sequence below:

Example:

Notice we are multiplying 2 by each term in the sequence above. If the pattern were to continue, the next term of the sequence above would be 64. This is a geometric sequence!

In this geometric sequence, it is easy for us to see what the next term is, but what if we wanted to know the 15^th term? Instead of writing out and multiplying our terms 15 times, we can use a shortcut, and that’s where the Geometric Sequence formula comes in handy!

Geometric Sequence Formula:

Take a look at the geometric sequence formula below, where each piece of our formula is identified with a purpose.

a_n=a₁r^(n-1)

a₁ = The first term is always going to be that initial term that starts our geometric sequence. In this case, our sequence is 4,8,16,32, …… so our first term is the number 4.

r= One key thing to notice about the formula below that is unique to geometric sequences is something called the Common Ratio. The common ratio is the number that is multiplied or divided to each consecutive term within the sequence.

n= Another interesting piece of our formula is the letter n, this always stands for the term number we are trying to find. A great way to remember this is by thinking of the term we are trying to find as the nth term, which is unknown.

Now that we broke down our geometric sequence formula, let’s try to answer our original question below:

Example #1: Common ratio r>1

Step 1: First let’s identify the common ratio between each previous and subsequent term of the sequence. Notice each term in the sequence is multiplied by 2 (as we identified earlier in this post). Therefore, our common ratio for this sequence is 2.

Step 2: Next, let’s write the geometric sequence formula and identify each part of our formula (First Term=4, Term number=15, common ratio=2).

Step 3: Now let’s fill in our formula and solve with the given values.

Let’s look at another example where, the common ratio is a bit different, and instead of multiplying a number, this time we are going to be dividing the same number from each subsequent term, (this can also be thought of as multiplying by a common ratio that is a fraction):

Example #2: Common ratio 0<r<1

Step 1: First let’s identify the common ratio between each number in the sequence. Notice each term in the sequence is divided by 2 (or multiplied by 1/2 that way it is shown below).

Step 2: Next, let’s write the geometric sequence formula and identify each part of our formula (First Term=1000, Term number=10, common ratio=1/2).

Step 3: Next let’s fill in our formula and solve with the given values.

Think you are ready to practice solving geometric sequences on your own? Try the following practice questions with solutions below:

Practice Questions:

Find the 12^th term given the following sequence: 1250, 625, 312.5, 156.25, 78.125, ….
Find the 17^th term given the following sequence: 3, 9, 27, 81, 243,…..
Find the 10^th term given the geometric sequence: 5000, 1250, 312.5, 78.125 …..
Shirley has $100 that she deposits in the bank. She continues to deposit twice the amount of money every month. How much money will she deposit in the twelfth month at the end of the year?

Solutions:

Fun Fact!

Did you know that the geometric sequence formula can be considered an explicit formula? An explicit formula means that even though we do not know the other terms of a sequence, we can still find the unknown value of any term within the given sequence. For example, in the first example we did in this post (example #1), we wanted to find the value of the 15th term of the sequence. We were able to do this by using the explicit geometric sequence formula, and most importantly, we were able to do this without finding the first 14 previous terms one by one…life is so much easier when there is an explicit geometric sequence formula in your life!

Other examples of explicit formulas can be found within the arithmetic sequence formula and the harmonic series.

Looking to learn more about sequences? You’ve come to the right place! Check out these sequence resources and posts below. Personally, I recommend looking at the finite geometric sequence or infinite geometric series posts next!

Arithmetic Sequence

Recursive Formula

Finite Arithmetic Series

Finite Geometric Series

Infinite Geometric Series

Golden Ratio in the Real World

Fibonacci Sequence

Still, got questions? No problem! Don’t hesitate to comment below or reach out via email. And if you would like to see more MathSux content, please help support us by following ad subscribing to one of our platforms. Thanks so much for stopping by and happy calculating!

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Rotations about a Point: Geometry

rotations about a point

Happy Wednesday math friends! In this post we’re going to dive into rotations about a point! In this post we will be rotating points, segments, and shapes, learn the difference between clockwise and counterclockwise rotations, derive rotation rules, and even use a protractor and ruler to find rotated points. The fun doesn’t end there though, check out the video and practice questions below for even more! And as always happy calculating! 🙂

What are Rotations?

Rotations are a type of transformation in geometry where we take a point, line, or shape and rotate it clockwise or counterclockwise, usually by 90º,180º, 270º, -90º, -180º, or -270º.

A positive degree rotation runs counter clockwise and a negative degree rotation runs clockwise. Let’s take a look at the difference in rotation types below and notice the different directions each rotation goes:

How do we rotate a shape?

There are a couple of ways to do this take a look at our choices below:

We can visualize the rotation or use tracing paper to map it out and rotate by hand.
Use a protractor and measure out the needed rotation.
Know the rotation rules mapped out below. Yes, it’s memorizing but if you need more options check out numbers 1 and 2 above!

Rotation Rules:

Where did these rules come from?

To derive our rotation rules, we can take a look at our first example, when we rotated triangle ABC 90º counterclockwise about the origin. If we compare our coordinate point for triangle ABC before and after the rotation we can see a pattern, check it out below:

The rotation rules above only apply to those being rotated about the origin (the point (0,0)) on the coordinate plane. But points, lines, and shapes can be rotates by any point (not just the origin)! When that happens, we need to use our protractor and/or knowledge of rotations to help us find the answer. Let’s take a look at the Examples below:

Example #1:

Step 1: First, let’s look at our point of rotation, notice it is not the origin we rotating about but point k! To understand where our triangle is in relation to point k, let’s draw an x and y axes starting at this point:

Step 2: Now let’s look at the coordinate point of our triangle, using our new axes that start at point k.

Step 2: Next, let’s take a look at our rule for rotating a coordinate -90º and apply it to our newly rotated triangles coordinates:

Step 3: Now let’s graph our newly found coordinate points for our new triangle .

Step 4: Finally let’s connect all our new coordinates to form our solution:

Another type of question with rotations, may not involve the coordinate plane at all! Let’s look at the next example:

Example #2:

Step 1: First, let’s identify the point we are rotating (Point M) and the point we are rotating about (Point K).

Step 2: Next we need to identify the direction of rotation. Since we are rotating Point M 90º, we know we are going to be rotating this point to the left in the clockwise direction.

Step 3: Now we can draw a line from the point of origin, Point K, to Point M.

Step 4: Now, using a protractor and ruler, measure out 90º, draw a line, and notice that point L lands on our 90º line. This is our solution! (Note: For help on how to use a protractor, check out the video above).

Ready for more? Check out the practice questions below to master your rotation skills!

Practice Questions:

Point B is rotated -90º about the origin. Which point represents newly rotated point B?

2. Triangle ABC is rotated -270º about point M. Show newly rotated triangle ABC as A prime B prime C prime.

3. Point G is rotated about point B by 180º. Which point represents newly rotated point B?

4. Segment AB is rotated 270º about point K. Show newly rotated segment AB.

Solutions:

Still got questions on how to rotations about a point? No problem! Don’t hesitate to comment with any questions or check out the video above for even more examples. Happy calculating! 🙂

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Looking for more Transformations? Check out the related posts below!

Translations

Dilations

Reflections

The Unit Circle: Algebra 2/Trig.

Greetings math friends! In today’s post we’re going to go over some unit circle basics. We will find the value of trigonometric functions by using the unit circle and our knowledge of special triangles. For even more practice questions and detailed info., don’t forget to check out the video and examples at the end of this post. Keep learning and happy calculating! 🙂

What is the Unit Circle?

The Unit Circle is a circle where each point is 1 unit away from the origin (0,0). We use it as a reference to help us find the value of trigonometric functions.

Notice the following things about the unit circle above:

Degrees follow a counter-clockwise pattern from 0 to 360 degrees.
Values of cosine are represented by x-coordinates.
Values of sine are represented by y-coordinates.
Using the unit circle we can find the degree and radian value of trigonometric functions (SOH CAH TOA). Check out the example below!

What’s the big deal with Quadrants?

Within a coordinate plane there are 4 quadrants numbered I, II, III, and IV used throughout all of mathematics. Within these quadrants there are different trigonometric functions that are positive to each unique quadrant. This will be important when solving questions with reference angles later in this post. Check out which trig functions are positive in each quadrant below:

Now let’s look at some examples on how to find trigonometric functions using our circle!

Negative Degree Values:

The unit circle also allows us to find negative degree values which run clockwise, check it out below!

Knowing that negative degrees run clockwise, we can now find the value of trigonometric functions with negative degree values.

How to find trig ratios with 30º, 45º and 60º ?

Instead of memorizing much, much more of the unit circle, there’s a trick to memorizing two simple special triangles for answering these types of questions. The 45º 45º 90º special triangle and the 30º 60º 90º special triangle. (Why does this work? These special triangles can also be derived and found on the unit circle).

Using the above triangles and some basic trigonometry in conjugation with the unit circle, we can find so many more angles, take a look at the example below:

Since we need to find the value of tan(45º) , we will use the 45º, 45º, 90º special triangle.

For our last question, we are going to need to combine our knowledge of unit circles and special triangles:

-> In order to do this, we must first look at where our angle falls on the unit circle. Notice that the angle 135º is encompassed by the pink lines and falls in quadrant 2.

-> Since our angle falls in the second quadrant where only the trig function sin is positive. Since we are finding an angle with the function cosine, we know the solution will be negative.

-> Now we need to find something called a reference angle. Which is what those θ, 180°-θ, θ-180°, 360°-θ and symbols represent towards the center of the unit circle. Using these symbols will help us find the value of cos(135º).

Because the angle we are trying to find,135º , falls in the second quadrant, that means we are going to use the reference angle that falls in that quadrant 180º-θ theta, using the angle we are given as θ.

-> Now we can re-write and solve our trig equations using our newly found reference angle, 45º.

Now we are going to use our 45º 45º 90º special triangle and SOH CAH TOA to evaluate our trig function. For a review on how to use SOH CAH TOA, check out this link here.

When you’re ready, try the problems on your own below!

Practice Questions:

Solve the following trig functions using a unit circle and your knowledge of special triangles:

Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above for even more examples. Happy calculating! 🙂

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Arithmetic Sequence Formula:

a_n=a₁+(n-1)d

a₁=First Term

n=Term Number in Sequence

d=Common Difference (Number Added/Subtracted to each Term in Sequence)

Hi everyone and welcome to Mathsux! In this post, we’re going to go over arithmetic sequences (otherwise known as arithmetic progression). We’ll identify what arithmetic sequences are, break down each part of the arithmetic sequence formula a_n=a₁+(n-1)d, and solve two different types of examples. As always if you want more questions, check out the video below and the practice problems at the end of this post. Happy calculating! 🙂

What are Arithmetic Sequences?

Arithmetic sequences are a sequence of numbers that form a pattern when the same number is either added or subtracted to each successive term. Take a look at the example of an arithmetic sequence below:

Notice the pattern? We are adding the number 2 to each term in the sequence above. If the pattern were to continue, the next term of the sequence above would be 10+2 which gives us 12. This is an arithmetic sequence!

In the above sequence, it’s easy for us to identify what the next term in the sequence would be, but what happens if we were asked to find the 123^rd term of an arithmetic sequence? That’s where the Arithmetic Sequence Formula would come in handy!

Arithmetic Sequence Formula:

Take a look at the arithmetic sequence formula below, where each piece of our formula is identified with a purpose.

a_n=a₁+(n-1)d

a₁= The first term is always going to be that initial term that starts our arithmetic sequence. In this case, our sequence is 4,6,8,10, …… so our first term is the number 4.

d = One key thing to notice about the formula below that is unique to arithmetic sequences is something called the Common Difference. The common difference is the number that is added or subtracted to each consecutive term within the sequence.

Now that we know the arithmetic sequence formula, let’s try to answer our original question below:

Step 1: First let’s identify the common difference between each previous and subsequent term of the sequence. Notice each term in the sequence is being added by 2 (like we identified earlier in this post). Therefore, our common difference for this sequence is 2.

Step 2: Next, let’s write the arithmetic sequence formula and identify each part of our formula (First Term=4, Term number=123, common difference=2).

Step 3: Fill in our formula and solve with the given values.

Now let’s look at another example where we subtract the same number from each term in the sequence, making the common difference negative.

Step 1: First let’s identify the common difference between each previous term and each subsequent term of the sequence. Notice each term in the sequence is being subtracted by 3. Therefore, our common difference for this sequence is -3, negative, because we are subtracting.

Step 2: Next, let’s write the arithmetic sequence formula and identify each part of our formula (First Term=100, Term number=12, common difference=-3).

Step 3: Finally, let’s fill in our formula and solve with the given values.

Think you are ready to practice solving arithmetic sequences on your own? Try the following practice questions with solutions below:

Practice Questions:

Find the 123^rd term given the following sequence: 8, 12, 16, 20, 24, ….
Find the 117^th term given the following sequence: 2, 2.5, 3, 3.5, …..
Find the 52^nd term given arithmetic sequence: 302, 300, 298, …..
A software engineer charges $100 for the first hour of consulting and $50 for each additional hour. How much would 500 hours of the consultation cost?

Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! 🙂

Fun Fact!

Did you know that the arithmetic sequence formula can be considered an explicit formula? An explicit formula means that even though we do not know the other terms of a sequence, we can still find the unknown value of any term within the given sequence. For example, in the first example we did in this post (example #1), we wanted to find the value of the 123rd term of the sequence. We were able to do this by using the explicit arithmetic sequence formula, and most importantly, we were able to do this without finding the first 122 previous terms one by one…life is so much easier when there is an explicit arithmetic sequence formula in your life!

Other examples of explicit formulas can be found within the geometric sequence formula and the harmonic series.

Looking to learn more about sequences? You’ve come to the right place! Check out these sequence resources and posts below. Personally, I recommend looking at the geometric sequence or finite arithmetic series posts next!

Geometric Sequence

Recursive Formula

Finite Arithmetic Series

Finite Geometric Series

Infinite Geometric Series

Golden Ratio in the Real World

Fibonacci Sequence

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45 45 90 Triangle

45 45 90 triangle

Greetings math folks! For anyone familiar with trigonometry and SOH CAH TOA trigonometric ratio you should know that there is something special about right triangles. We are about to learn more about right triangles, as there are two distinct types of special right triangles in this world that we need to know, this includes the 45 45 90 triangle and the 30 60 90 triangle. In this post, we are going to go over the 45 45 90 special right triangle! If you are looking for the other very famous special triangle, (30 60 90), check out this post here.

With the help of this special triangle, we are going to see how to find the missing sides of a right triangle when given only one of its lengths (and the angles of the right triangle given are 45 45 90). For even more examples, check out the video and practice questions below and at the end of this post. Happy calculating! 🙂

45 45 90 Right Triangle Ratio:

Looking above at our 45 45 90 special triangle, notice it contains one right angle and 2 equal angles of 45 degrees. Based on these angle proportions, we are able to infer information about each sides length, thats where our ratio comes in!

45 45 90 Triangle – Why is it special? Where did come from?

45 45 90 special right triangles are “special” because they are a type of Isosceles Right Triangle, meaning they have two equal sides (marked in blue below). If we know that the isosceles triangle has two equal lengths, we can find the value of the length of the hypotenuse by using the Pythagorean Theorem based on the other two equal sides. Check out how we derive the 45 45 90 ratio below!

Notice, we started with the Pythagorean Theorem, then filled in our variables based on each given sides length. Next, we combined like terms and then took the square root of each side of the equation. Lastly, we found the value of hypotenuse, c, based on the other two legs, which is equal to the length of a times radical 2.

Now that we know the length of the hypotenuse in terms of each sides length a, we can re-label our triangle. Since we found the length of the hypotenuse in relation to the two equal legs, notice that this creates a ratio that applies to each and every triangle out there!

How do I use this ratio?

Ok, great we have derived the 45 45 90 ratio, but what do I do with this thing and how do I use it?

Knowing the above ratio, allows us to find the length of any missing side of a 45 45 90 special triangle (when given the value of one of its sides).

Let’s now try some 45 45 90 right triangle examples with missing sides below:

Example #1:

Step 1: First, let’s look at our ratio and compare it to our given right triangle.

Step 2: Notice we are given the value of the bottom leg, a=8. Knowing this we can fill in each length of our right triangle based on the ratio of a 45 45 90 special triangle shown below:

Now let’s look at an example where we are given the length of the hypotenuse and need to find the values of the other two missing sides of a 45 45 90 right triangle.

Example #2:

Step 1: First, let’s look at our ratio and compare it to our given right triangle.

Step 2: In this case, we need to do a little math to find the value of a, based on the Pythagorean Theorem. See how we use the Pythagorean Theorem step by step below to find the value of missing sides represented by a.

Write out the Pythagorean Theorem Formula
Fill in the values from our given 45 45 90 triangle based on the side lengths
Combine like terms a2 + a2 = 2a2 given.
Take the square root of both sides of the equation
Divide both sides by radical 2 to get a alone
Rationalize the denominator by multiplying the numerator and denominator by radical 2 and simplify

We have found our solution!

Now try mastering the art of the 45 45 90 special triangle on your own with the practice problems below!

Practice Questions:

Find the value of each missing side length of each 45 45 90 right triangle.

Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! 🙂

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Synthetic Division and Factoring Polynomials: Algebra 2/Trig.

Hey there math friends! In this post we will go over how and when to use synthetic division to factor polynomials! So far, in algebra we have gotten used to factoring polynomials with variables raised to the second power, but this post explores how to factor polynomials with variables raised to the third degree and beyond!

If you have any questions don’t hesitate to comment or check out the video below. Also, don’t forget to master your skills with the practice questions at the end of this post. Happy calculating! 🙂

What is Synthetic Division?

Synthetic Division is a shortcut that allows us to easily divide polynomials as opposed to using the long division method. We can only use synthetic division when we divide a polynomial by a binomial in the form of (x-c), where c is a constant number.

Example #1:

*Notice we can use synthetic division in this case because we are dividing by (x+4) which follows our parameters (x-c), where c is equal to 4.

Example #2: Factoring Polynomials

Let’s take a look at the following example and use synthetic division to factor the given polynomial:

Check!

The great thing about these questions is that we can always check our work! If we wanted to check our answer, we could simply distribute 2(x+1)(x+3)(x-2) and get our original polynomial, f(x)=2x³+4x²-10x-12.

Try the practice problems on your own below!

Looking to brush up on how to divide polynomials the long way using long division? Check out this post here!

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! 🙂

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Variance and Standard Deviation: Statistics

Greetings math friends! In this post, we arere going to go over the formulas for Variance and Standard Deviation. We will take this step by step by explaining the significance of the variance and standard deviation formulas in relation to a set of data. Get your calculators ready because this step by step although not hard, will take some serious number crunching! Also, don’t forget to check out the video on standard deviation and variance below to see how to check your work using a calculator. Happy calculating! 🙂

If you’re looking for related formulas, Mean Absolute Deviation (MAD) and Expected Value, scroll to the bottom of this post! And if you’re interested we’ll also touch upon the difference between population variance and sample variance later in this post.

What is the Variance?

The variance represents the spread of data or the distance each data point is from the mean. When we have multiple observations in our data, we want to know how far each unit of data is from the mean. Are all the data points close together or spread far apart? What is the probability distribution? This is what the variance will help tell us!

Don’t freak out but here’s the formula for variance, notated as using the greek letter, sigma squared, σ²:

where…

x_i= Value of Data Point

μ= mean

n=Total Number of Data Points]

(x_i-μ)=Distance each data point is to the mean

In plain English, this translates to:

Let’s try an example to find the standard deviation and variance of the data set below.

Step 1: First, let’s find the mean, μ.

Step 2: Now that we have the mean, we are going to do each part of our formula one step at a time in the table below.

Notice we subtract each test score from the mean, μ=78. Then we square the result of each subtracted test score to get the squared deviation of each data value, then finally sum all the squared results together.

Step 3: Now that we summed all of our squared deviations, to get 730, we can fill this in as our numerator in the variance formula. We also know our denominator is equal to 5 because that is the total number of test scores in our data set.

What is Standard Deviation?

Standard deviation is a unit of measurement that is unique to each data set and is used to measure the spread of data. The standard deviation formula happens to be very similar to the variance formula!

Below is the formula for standard deviation, notated as sigma, the greek letter, σ:

Since this is the same exact formula as variance with a square root, all we need to do is take the square root of the variance to find standard deviation:

Sample VS. Population

What is the difference between a sample vs. a population?

A population in statistics refers to an entire data set that at times can be humanly incapable of reaching.

For example: If we wanted to know the average income of everyone who lives in New York State, it would be almost impossible to reach every working person and ask them how much they make for a living.

To make up for the impossibility of data collection, we usually only survey a sample of the entire population to get income levels of let’s say 10,000 people across New York State, a much more reasonable in terms of data collecting!

And taking this sample size from the entire working population of New York State provides us with a sample mean, a sample variance, and a sample standard deviation.

On the other hand, if we were able to ask every student in a school what their grade point average was and get an answer, this would be an example of a whole population. Using this information, we would be able to find the population mean, population variance, and population standard deviation.

Sample notation also differs from population notation, but don’t worry about these too much, because the formulas and meanings remain the same. For example, the population mean is represented by the greek letter, μ, but the sample mean is represented by x bar.

Now try calculating the standard deviation of each data set below on your own with the following practice problems!

Practice Questions:

Solutions:

Other Related Formulas

Mean Absolute Deviation (MAD):

The Mean Absolute Deviation otherwise known as MAD is another formula related to variance and standard deviation. In the MAD formula above, notice we are doing very similar steps, by finding the distance to the mean of each data point, only this time we are taking the aboslute value of the ditsance to the mean. Then we sum all the absolute value distances together and divide by the total number of data points.

Why do we use aboslute value in this formula? We take the absolut value, because if didn’t the distance to the means summed togther would cancel eachother out to get zero!

Where…

X = Data point value

μ = mean

N=Total number of data points

|X-μ|=absolute deviation

If we were to take the sample from our example earlier,60, 85, 95, 70, 80, in this post and find the MAD it would go something like this:

Expected Value:

Expected Value is the weighted average of all possible outcomes of one “game” or “gamble” based on the respective probabilities of each potential outcome for a discrete random variable. A “gamble” is defined by the following rules: 1) All possible outcomes are known 2) An outcome cannot be predicted 3) All possible outcomes are of numeric value and 4) The Game can be repeated multiple times under the same conditions.

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! 🙂

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Also! If you’re looking for more statistics, check out this post on how to create and analyze box and whisker plots here!

Combining Like Terms and Distributive Property: Algebra

Greetings math peeps! In today’s post, we are going to review some of the basics: combining like terms and distributive property. It’s so important to master the basics such as these, so you’re prepared and ready to handle the harder stuff that’s just around the corner, trust me they’re coming! And for those who already feel comfortable with these topics, great! Skip ahead and try the practice questions at the bottom of this post and happy calculating! 🙂

When do we combine “like terms?”

Combining like terms allows us to simplify and calculate our answer with terms that have the same variable and same exponent values only. For example, we can combine the following expression:

distributive property and combining like terms

How do we combine like terms?

We add or subtract the whole number coefficients and keep the variable they have in common.

Why? We could not add these two terms together because their variables do not match! 2 is multiplied by x, while 3 is multiplied by the variable xy.

Why? We could not add these two terms together because their variables and exponents do not match! 2 is multiplied by x, while 3 is multiplied by the variable x^2 . Exponents for each variable must match to be considered like terms.

Distributive Property:

Combining like terms and the distributive property go hand in hand. The distributive property rule states the following:

There are no like terms to combine in the example above, but let’s see what it would like to use the distributive property and combine like terms at the same time with the following examples:

Example #1:

Example #2:

In some cases, we also have to distribute the -1 that can sometimes “hide” behind a parenthesis.

Try the following questions on your own on combining like terms and the distributive property and check out the video above for more!

Practice Questions:

Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! 🙂

Looking to review more of the basics? Check out this post on graphing equations of a line y=mx+b here.

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Transversals and Parallel Lines: Geometry

Happy Wednesday math friends! In this post we are going to look at transversals and parallel lines and find the oh so many congruent and supplementary angles they form when they come together! Congruent angles that form with these types of lines are more commonly known as Alternate Interior Angles, Alternate Exterior Angles, Corresponding angles, and Supplementary angles. Let’s look at this one step at a time:

What are Transversals and Parallel Lines?

When two parallel lines are cut by a diagonal line ( called a transversal) it looks something like this:

Each angle above has at least one congruent counterpart. There are several different types of congruent relationships that happen when a transversal cuts two parallel lines and we are going to break each down:

1) Alternate Interior Angles:

When a transversal line cuts across two parallel lines, opposite interior angles are congruent.

2) Alternate Exterior Angles:

When a transversal line cuts across two parallel lines, opposite exterior angles are congruent.

3) Corresponding Angles:

When a transversal line cuts across two parallel lines, corresponding angles are congruent.

4) Supplementary Angles:

Supplementary angles are a pair of angles that add to 180 degrees. 180 degrees is the value of distance found within a straight line, which is why you’ll find so many supplementary angles below:

Knowing the different sets of congruent and supplementary angles, we can easily find any missing angle values when faced with the following question:

-> Using our knowledge of congruent and supplementary angles we should be able to figure this out! Right away we can find angle 2 by noticing angle 1 and angle 2 are supplementary angles (add to 180 degrees).

-> Knowing angle 2 is 50 degrees, we can now fill in the rest of our transversal angles based on our corresponding and supplementary rules.

Try the following transversal and parallel lines questions below! Some may a bit harder than the previous example, if you get stuck, check out the video that goes over a similar example above and happy calculating! 🙂

Practice Questions:

Find the value of the missing angles given line r is parallel to line s and line t is a transversal.

2. Find the value of the missing angles given line r is parallel to line s and line t is a transversal.

Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! 🙂

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Also, if you’re looking to learn about the difference between parallel and perpendicular lines, check out this post here!

Imaginary and Complex Numbers: Algebra 2/Trig.

Happy Wednesday and back to school season math friends! This post introduces imaginary and complex numbers when raised to any power exponent and when multiplied together as a binomial. When it comes to all types of learners, we got you between the video, blog post, and practice problems below. Happy calculating! 🙂

What are Imaginary Numbers?

Imaginary numbers happen when there is a negative under a radical and looks something like this:

Why does this work?

In math, we cannot have a negative under a radical because the number under the square root represents a number times itself, which will always give us a positive number.

Example:

But wait, there’s more:

When raised to a power, imaginary numbers can have the following different values:

Knowing these rules, we can evaluate imaginary numbers, that are raised to any value exponent! Take a look below:

-> We use long division, and divide our exponent value 54, by 4.

-> Now take the value of the remainder, which is 2, and replace our original exponent. Then evaluate the new value of the exponent based on our rules.

What are Complex Numbers?

Complex numbers combine imaginary numbers and real numbers within one expression in a+bi form. For example, (3+2i) is a complex number. Let’s evaluate a binomial multiplying two complex numbers together and see what happens:

-> There are several ways to multiply these complex numbers together. To make it easy, I’m going to show the Box method below:

Try mastering imaginary and complex numbers on your own with the questions below!

Practice:

Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Don’t forget to sign up for FREE weekly MathSux videos, lessons, and practice questions. Thanks for stopping by and happy calculating! 🙂

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Also, if you’re looking to learn more about dividing polynomials, check out this post here!

Looking for more on Quadratic Equations and functions? Check out the following Related posts!

Factoring Review

Factor by Grouping

Completing the Square

The Discriminant

Is it a Function?

Focus and Directrix of a Parabola

Quadratic Equations with 2 Imaginary Solutions

What are Geometric Sequences?

Geometric Sequence Formula:

Example #1: Common ratio r>1

Example #2: Common ratio 0<r<1

Practice Questions:

Solutions:

Fun Fact!

Related Posts:

What are Rotations?

How do we rotate a shape?

Rotation Rules:

Where did these rules come from?

Example #1:

Example #2:

Practice Questions:

Solutions:

What is the Unit Circle?

What’s the big deal with Quadrants?

Negative Degree Values:

How to find trig ratios with 30º, 45º and 60º ?

Practice Questions:

Solutions:

Related Trigonometry Posts:

What are Arithmetic Sequences?

Arithmetic Sequence Formula:

Practice Questions:

Solutions:

Fun Fact!

Related Posts:

45 45 90 Right Triangle Ratio:

45 45 90 Triangle – Why is it special? Where did come from?

How do I use this ratio?

Example #1:

Example #2:

Practice Questions:

Solutions:

Related Trigonometry Posts:

What is Synthetic Division?

Example #1:

Example #2: Factoring Polynomials

What is the Variance?

What is Standard Deviation?

Sample VS. Population

Practice Questions:

Solutions:

Other Related Formulas

Mean Absolute Deviation (MAD):

Expected Value:

When do we combine “like terms?”

How do we combine like terms?

Distributive Property:

Example #1:

Example #2:

Practice Questions:

Solutions:

What are Transversals and Parallel Lines?

1) Alternate Interior Angles:

2) Alternate Exterior Angles:

3) Corresponding Angles:

4) Supplementary Angles: