How to Solve Log Equations: Algebra 2/Trig.

Welcome to Mathsux! Today, we’re going to go over how to solve log equations, yay! But before we get into finding x, though, we need to go over what log equations are and why we use them in the first place…..just in case you were curious!

Also, if you have any questions about anything here, don’t hesitate to comment below or shoot me an email.  Happy calculating! 🙂

What are Log Equations?

Logarithms are the inverses of exponential functions.  This means that when graphed, they are symmetrical along the line y=x.  Check it out below!

How to Solve Log Equations

When on the same set of axis, notice how the functions are symmetrical over the line y=x:

How to Solve Log Equations

We use logarithms to find the unknown values of exponents, such as the x value in the equation, Screen Shot 2020-06-24 at 9.30.55 PM.png.  This is a simple example, where we know the value of x is equal to 2,(Screen Shot 2020-06-24 at 9.32.30 PM.png). But what if it were to get more complicated?  That’s where logs come in!

How to Solve Log Equations?

Logarithms follow a swooping pattern that allows us to write it in exponential form, let’s take a look at some Examples below:

How to Solve Log Equations

But wait there’s more! Logs have a set of Rules that makes solving log equations a breeze!

How to Solve Log Equations

We can use these rules to help us algebraically solve logarithmic equations, let’s look at an example that applies the Product Rule.

Example:

Screen Shot 2020-06-24 at 9.36.08 PM.png
Screen Shot 2020-06-24 at 9.36.50 PM
Screen Shot 2020-06-24 at 9.46.07 PM.png
Screen Shot 2020-06-24 at 9.38.32 PM

Try the following practice questions on your own!

Practice Questions:

Screen Shot 2020-06-24 at 9.39.16 PM.png

Solutions:

Screen Shot 2020-06-24 at 9.40.37 PM

Still got questions?  No problem! Check out the video that goes over the same example outlined above.  And for more info. on logarithms check out this post that goes over a NYS Regent’s question here.  Subscribe below to get the latest FREE math videos, lessons, and practice questions from MathSux. Thanks for stopping by and happy calculating! 🙂

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****Check out this Bonus Video on How to Change Log Bases****

How to Graph Equation of a Line, y=mx+b: Algebra

Hi everyone, welcome back to Mathsux! This week we’ll be reviewing how to graph an equation of a line in y=mx+b form. And if you have not checked out the video below, please do! Happy calculating! 🙂

how to graph y=mx+b

An equation of a line can be represented by the following formula:

y=mx+b

Y-Intercept: This is represented by b, the stand-alone number in y=mx+b. This represents where the line hits the y-axis.  This is always the first point you want to start with when graphing at coordinate point (0,b).

Slope: This is represented by m, the number next to x in y=mx+b. Slope tells us how much we go up or down the y-axis and left or right on the x- axis in fraction form:

how to graph equation of a line

Now let’s check out an Example!

Graph the following:

Screen Shot 2020-06-17 at 9.10.42 PM

-> First, let’s identify the slope and y-intercept of our line.

how to graph equation of a line

-> To start, let’s graph the first point on our graph, the y-intercept at point (0,1):

how to graph equation of a line

-> Now for the slope. We are going to go up one and over to the right one for each point, since our slope is 1/1.

how to graph equation of a line

-> Connect all of our coordinate points and label our graph.

how to graph equation of a line

Try the following practice questions on your own!

Practice Questions:

how to graph equation of a line
how to graph equation of a line

Want more Mathsux?  Don’t forget to check out our Youtube channel and more below! And if you have any questions, please don’t hesitate to comment below. Happy Calculating! 🙂

Need to brush up on slope? Click here to see how to find the rate of change.

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Perpendicular & Parallel Lines Through a Given Point: Geometry

Happy Wednesday math friends! Today we’re going to go over the difference between perpendicular and parallel lines, then we’ll use our knowledge of the equation of a line (y=mx+b) to see how to find perpendicular and parallel lines through a given point.  This is a common question that comes up on the NYS Geometry Regents and is something we should prepare for, so let’s go!

If you need any further explanation, don’t hesitate to check out the Youtube video below that goes into detail on how to find perpendicular and parallel lines through a given point one step at a time. Happy calculating! 🙂

Perpendicular Lines:

Perpendicular & Parallel Lines Through a Given Point

Perpendicular Lines: Lines that intersect to create a 90-degree angle and can look something like the graph below.  Their slopes are negative reciprocals of each other which means they are flipped and negated. See below for example!

Example: Find an equation of a line that passes through the point (1,3) and is perpendicular to line y=2x+1 .

Screen Shot 2020-06-10 at 10.28.20 AM
Perpendicular & Parallel Lines Through a Given Point
Perpendicular & Parallel Lines Through a Given Point
Screen Shot 2020-06-10 at 10.29.06 AM

Parallel Lines:

Parallel lines are lines that go in the same direction and have the same slope (but have different y-intercepts). Check out the example below!

Perpendicular & Parallel Lines Through a Given Point

Example: Find an equation of a line that goes through the point (-5,1) and is parallel to line y=4x+2.

Screen Shot 2020-06-10 at 10.34.46 AM
Screen Shot 2020-06-10 at 10.35.23 AM

Try the following practice questions on your own!

Practice Questions:

1) Find an equation of a line that passes through the point (2,5) and is perpendicular to line y=2x+1.

 2) Find an equation of a line that goes through the point (-2,4) and is perpendicular to lineScreen Shot 2020-06-10 at 11.24.06 AM

 3)  Find an equation of a line that goes through the point (1,6) and is parallel to line y=3x+2.

4)  Find an equation of a line that goes through the point (-2,-2)  and is parallel to line y=2x+1.

Solutions:

Screen Shot 2020-06-10 at 11.22.05 AM

Need more of an explanation? Check out the video that goes over these types of questions up on Youtube (video at top of post) and let me know if you have still any questions.

Happy Calculating! 🙂

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Looking for more on Perpendicular and parallel lines? Check out this Regents question on perpendicular lines here!

4 Ways to Factor Trinomials: Algebra

Greeting math peeps and welcome to MathSux! In this post, we are going to go over 4 ways to Factor Trinomials and get the same answer, including, (1) Quadratic Formula (2) Product/Sum, (3) Completing the Square, and (4) Graphing on a Calculator.  If you’re looking for more don’t forget to check out the video and practice questions below.  Happy Calculating! 🙂

Also, if need a review on Factor by Grouping or Difference of Two Squares (DOTS) check out the hyperlinks here!

*If you haven’t done so, check out the video that goes over this exact problem, and don’t forget to subscribe!

We will take this step by step, showing 4 ways to factor trinomials, getting the same answer each and every time! Let’s get to it!

4 Ways to Factor Trinomials

Screen Shot 2020-06-02 at 3.03.55 PM

(1) Quadratic Formula:

4 Ways to Factor Trinomials

____________________________________________________________________

(2) Product/Sum:

4 Ways to Factor Trinomials____________________________________________________________________

(3) Completing the Square:

4 Ways to Factor Trinomials____________________________________________________________________

(4) Graph:

4 Ways to Factor Trinomials

Choose the factoring method that works best for you and try the practice problems on your own below!

Practice Questions:

Screen Shot 2020-06-02 at 3.09.58 PM

Solutions:

Screen Shot 2020-06-02 at 3.10.30 PM

Want a review of all the different factoring methods out there?  Check out the ones left out here (DOTS and GCF) and happy calculating! 🙂

For even more ways to factor quadratic equations, check out How to factor by Grouping here! 🙂

Also, if you want more Mathsux?  Don’t forget to check out our Youtube channel and more below! If you have any questions, please don’t hesitate to comment below. Happy Calculating! 🙂

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Median of a Trapezoid Theorem: Geometry

Hi everyone and welcome to Math Sux! In this post, we are going to look at how to use and applythe median of a trapezoid theorem. Thankfully, it is not a scary formula, and one we can easily master with a dose of algebra. The only hard part remaining, is remembering this thing! Take a look below to see a step by step tutorial on how to use the median of a trapezoid theorem and check out the practice questions at the end of this post to truly master the topic. Happy calculating! 🙂

*If you haven’t done so, check out the video that goes over this exact problem, also please don’t forget to subscribe!

Medians of a Trapezoid copy
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Step 1:  Let’s apply the Median of a Trapezoid Theorem to this question!  A little rusty?  No problem, check out the Theorem below.

Median of a Trapezoid Theorem

Median of a Trapezoid Theorem: The median of a trapezoid is equal to the sum of both bases.Step 2: Now that we found the value of x , we can plug it back into the equation for Screen Shot 2020-06-02 at 7.33.44 AMmedian,  to find the value of median Screen Shot 2020-06-02 at 7.34.25 AM

Screen Shot 2020-06-02 at 7.34.48 AM

Want more practice?  Your wish is my command! Check out the practice problems below:

Practice Questions:

Median of a Trapezoid Theorem
Median of a Trapezoid Theorem
Median of a Trapezoid Theorem

1.Screen Shot 2020-06-02 at 7.35.29 AMis the median of trapezoid ABCDEF, find the value of the median, given the following:2. Screen Shot 2020-06-02 at 9.01.08 AMis the median of trapezoid ACTIVE, find the value of the median, given the following:3.Screen Shot 2020-06-02 at 9.17.01 AMis the median of  trapezoid DRAGON, find the value of the median, given the following:

Median of a Trapezoid Theorem

4. Screen Shot 2020-06-02 at 9.23.08 AMis the median of trapezoid MATRIX, find the value of the median, given the following:

Solutions:

Screen Shot 2020-06-02 at 9.25.05 AM

Need more of an explanation?  Check out the detailed video and practice problems. Happy calculating! 🙂

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Completing the Square: Algebra

Want to learn the ins and out of completing the square?  Then you’ve come to the right place! Learn how to Complete the Square step by step in the video and article below, then try the practice problems at the end of this post to truly master the topic! If you’re looking for more on completing the square, check out this post here. Happy Calculating! 🙂 Check out the video below for an in-depth look at completing the square:
Completing the square To answer this question, there are several steps we must follow including: Step 1: Move the whole number, which in this case is 16, to the other side of the equation.This image has an empty alt attribute; its file name is Screen-Shot-2020-12-25-at-6.07.43-PM.png This image has an empty alt attribute; its file name is Screen-Shot-2020-12-25-at-6.08.19-PM.png Step 2: Make space for our new number on both sides of the equation.  This number is going to be found by using a particular formula shown below: Completing the square Step 3: Add the number 9 to both sides of the equation, which we found using our formula. Completing the square Step 4: Combine like terms on the right side of the equation, adding 16+9 to get 25. Completing the square Step 5: Now, we need to re-write the left side of the equation using the following formula. Completing the square Step 6: Finally, we solve for x by taking the positive and negative square root to get the following answer and solve for two different equations:  Completing the square This image has an empty alt attribute; its file name is Screen-Shot-2020-12-25-at-6.29.22-PM.png Practice Questions: completing the square Solutions: completing the square

Want more Mathsux?  Don’t forget to check out our Youtube channel and more below! And if you have any questions, please don’t hesitate to comment below. Happy Calculating! 🙂

Need more of an explanation?  Check out why we complete the square in the first place here ! 🙂

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Area of a Sector: Geometry

Youtube Area of a sector copy

Hi math friends, has anyone been cooking more during quarantine?  We all know there is more time for cookin’ and eatin’ cakes but have you ever been curious about the exact amount of cake you are actually eating?! Well, you’re in luck because today we are going to go over how to find the area of a piece of cake, otherwise known as the Area of a Sector!

Now, we’ll all be able to calculate just how much we are overdoing it on that pie! Hopefully, everyone is eating better than I am (I should really calm down on the cupcakes).  Ok, now to our question:

*Also, If you haven’t done so, check out the video that goes over this exact problem, and don’t forget to subscribe!

Screen Shot 2020-05-19 at 4.18.42 PM

Explanation:

How do I answer this question? 

We must apply/adjust the formula for the area of a circle to find the area of the blue shaded region otherwise known as the sector of this circle.                                                    

How do we do this?    

Before we begin let’s review the formula for the area of a circle. Just a quick reminder of what each piece of the formula represents:

area of a sector

Step 1: Now let’s fill in our formula, we know the radius is 5, so let’s fill that in below:

Screen Shot 2020-05-19 at 4.26.08 PM

Step 2: Ok, great! But wait, this is for a sector; We need only a piece of the circle, not the whole thing.  In other words, we need a fraction of the circle. How can we represent the area of the shaded region as a fraction?

Well, we can use the given central angle value, Screen Shot 2020-05-19 at 4.27.17 PM, and place it over the whole value of the circle,Screen Shot 2020-05-21 at 4.01.12 PM . Then multiply that by the area of the entire circle. This will give us the value we are looking for!

area of a sector
area of a sector

Step 3: Multiply and solve!

Ready for more? Try solving these next few examples on your own to truly master area of a sector!

Practice Questions:

Find the area of each shaded region given the central angle and radius for each circle:

area of a sector

Solutions:

Screen Shot 2020-05-19 at 4.30.36 PM

What do you think of finding the area of sector? Are you going to measure the area of your next slice of pizza?  Do you have any recipes to recommend?  Let me know in the comments and happy calculating! 🙂

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Looking for more about circles? Check out this post on the circle formula here!

Solving Radical Equations: Algebra 2/Trig.

Screen Shot 2020-05-11 at 9.01.41 PM

Today we’re back with Algebra 2, this time solving for radical equations!  Did you say “Radical Equations?” As in wild and crazy equations? No, not exactly, radicals in math are used to take the square root, cubed root, or whatever root of a number.

Solving Radical Equations
Solving Radical Equations

Example #1:

Screen Shot 2020-05-12 at 11.25.03 AM.png

Radicals are actually pretty cool because we can write them a couple of different ways and they all mean the same thing! Check it out below:Still not sure of their coolness? Let’s see what they look like with actual numbers:
Example: Solve the following algebraic equation below for the missing variable (aka, solve for x).Explanation:

How do I answer this question?   

The question wants us to solve for x using our knowledge of radicals and algebra. You can also check out how to solve this question on Youtube here!

How do we do this? 

Step 1: We start solving this radical equation like any other algebraic problem: by getting x alone. We can do this easily by subtracting 7 and then dividing out 5.

Solving Radical Equations

Step 2: Now, to get rid of that pesky radical, we need to square the entire radical.  Remember, whatever we do to one side of the equation, we must also do to the other side of the equation, therefore, we also square 14 on the other side of the equal sign. *This gets rid of our radical and allows us to solve for x algebraically as normal!

Solving Radical Equations
Screen Shot 2020-05-12 at 11.29.34 AM.png

What happens when there is a cubed root though!?!?

When dividing polynomials with different value roots, raise the entire radical to that same power of root to cancel it out:Remember, we know radicals can also be written as fractions:

Solving Radical Equations

Therefore we also know that if we raise the entire radical expression to the same power of the root, the two exponents will cancel each other out:

Solving Radical Equations

Example #2:

Solving Radical Equations

Want more practice? Try solving radical equations in the next few examples on your own. 

Practice:

Screen Shot 2020-05-12 at 11.32.39 AM.png

Solutions:

Screen Shot 2020-05-12 at 11.33.12 AM.png

Looking to brush up on how to solve absolute value equations? Check out the post here! Did I miss anything?  Don’t let any questions go unchecked and let me know in the comments! Happy calculating! 🙂 

Don’t forget to check out the latest free videos and posts with MathSux and subscribe!

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Dividing Polynomials: Algebra 2/Trig.

Greeting math peeps! In this post we are going to go over dividing polynomials! At some point, you may need to know how to answer these types of questions. The cool thing about dividing polynomials is that it’s the same long division you did way back in grade school (except now with a lot of x). Ok, let’s get to it and check out the question below:

Also, if you haven’t done so, check out the video related that corresponds to this problem on Youtube below! 🙂

Screen Shot 2020-05-04 at 10.21.17 PM

Explanation:

How do I answer this question?

The question wants us to divide polynomials by using long division.

How do we do this?     

Step 1: First we set up a good ole’ division problem with the divisor, dividend, and quotient to solve.

Dividing Polynomials

Step 2: Now we use long division like we used to back in the day! If you have any confusion about this please check out the video in this post.

Dividing Polynomials
Dividing Polynomials

What if there’s a Remainder?

What happens when there is a remainder though!?!? When dividing polynomials with a remainder in the quotient, the answer is found and checked in a very similar way! Check it out in the example below:

Screen Shot 2020-05-04 at 10.23.44 PM
Dividing Polynomials

Notice we represented the remainder by adding Screen Shot 2020-05-04 at 10.35.06 PM to our quotient! We just put the remainder over the divisor to represent this extra bit of solution.

Want more practice? Try solving these next few examples on your own. 

Practice Questions:

Screen Shot 2020-05-04 at 10.45.10 PM.png

Solutions:

Screen Shot 2020-05-04 at 10.45.37 PM

If you’re looking for more on dividing polynomials, check out this post on synthetic division and finding zeros 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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Earth Day Fractals!

In honor of Earth Day last week, I thought we’d take a look at some math that appears magically in nature.  Math? In nature?  For those of you who think math is unnatural or just terrible in general, this is a great time to be proven otherwise!

The key that links math to nature is all about PATTERNS. All math is based on is patterns.  This includes all types of math, from sequences to finding x, each mathematical procedure follows some type of pattern. Meanwhile back in the nearest forest, patterns are occurring everywhere in nature.

The rock star of all patterns would have to be FRACTALS. A Fractal is a repeating pattern that is ongoing and has different sizes of the exact same thing!  And the amazing thing is that we can actually find fractals in our neighbor’s local garden.

Let’s look at some Fractal Examples:

(1) Romanesco Broccoli:  Check out those repeating shapes, that have the same repeating shapes on those shapes and the same repeating shapes on even smaller shapes and…. my brain hurts!

Screen Shot 2020-04-26 at 10.18.47 PM                                                          Screen Shot 2020-04-30 at 10.15.34 PM

(2) Fern Leaves:  The largest part of this fractal is the entire fern leaf itself.  The next smaller and identical part is each individual “leaf” along the stem.  If you look closely, the pattern continues!

Screen Shot 2020-04-26 at 10.30.02 PM                                                        Screen Shot 2020-04-30 at 10.16.13 PM

(3) Leaves:  If you’ve ever gotten up real close to any type of leaf, you may have noticed the forever repeating pattern that gets smaller and smaller. Behold the power and fractal pattern of this mighty leaf below!

Screen Shot 2020-04-27 at 3.45.36 PM.                                                        Screen Shot 2020-04-30 at 10.16.55 PM

Just in case fractals are still a bit hard to grasp, check out the most famous Fractal below,  otherwise known as Sierpinski’s Triangle.  This example might not be found in your local back yard, but it’s the best way to see what a fractal truly is up close and infinite and stuff.

fractals
Screen Shot 2020-04-30 at 10.19.21 PM

Looking for more math in nature?  Check out this post on the Golden Ratio and happy calculating! 🙂

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