How to Solve Quadratic Equations by Factoring

Kori Vagner By :
September 2, 2017 in , , , , ,

Algebra Quadratic Equations

In order to learn how to solve a quadratic equation by factoring, you first need to know how to factor quadratics. Do you remember how to do this? If not, no big deal. I will briefly review this for you.

So what exactly is a quadratic? A quadratic is a polynomial that looks like this: x^2+12x+36

The first thing you want to ask yourself is: What 2 numbers will add up to the middle number in the equation (12) and what 2 numbers will ALSO multiply to get the last number in the equation (36).

The best advice I can give you is just try and think of numbers that multiply together to get 36 and once you find those numbers, try adding them together to see if they add to get the middle number, 12……..

Do you think you have it, yet?

The correct answer is 6 and 6. 6×6=36 and 6+6= 12. Easy, right? Next, you will set the numbers within the parenthesis (x+6)(x+6).They both have a positive sign because I was multiplying to get a positive 36.

**If you had a positive 36 and a negative 12, your answer would be (x-6)(x-6). Because -6 x-6=36 and -6+-6= -12 Pay attention to the signs in the quadratic!

LET’S DIVE IN

Okay, hopefully factoring quadratics is coming back to you! Now, let’s move on! The new part to this, is that the quadratic expression is now part of the equation. You will be solving for the values of the variable that make the equation true.

For example, if you have the quadratic expression: x^2-9x+20 and you factor to get:
(x-4)(x-5), you are not quite done!!

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In order to actually solve this equation, you have to understand the “Zero Product Property Rule.” All this says is that we can ONLY come to a conclusion of the answer if the product itself is equal to zero. If for some reason the product of factors is equal to a non-zero number, then we can not draw a conclusion to the values of the factors.

Now that you understand the Zero Product Property Rule, you must always have the equation in the form of “quadratic expression” = “0” before making any attempt to solve the equation by factoring!

So going back to the original example, x^2-9x+20, when we factor to get (x-4)(x-5), one of the factors MUST be equal to zero, so you will need to set both factors to zero, which will look as follows:

(x-4)=0
or
(x-5)=0

The solution will need to read: x=4 and x=5 or x=4,5 (Either way is fine. They both mean the same thing) NOTE: You will notice that x DOES NOT = -4 and -5. You always change the signs, in order to get your equation to equal to zero!

Now, let’s check to see if our original equation equals zero!

Original equation: x^2-9x+20

Factored: (x-4)(x-5)=0

Solution: x=4 or x=5

CHECK (x=4):

4^2-9(4)+20
=16-36+20
=0

CHECK (x=5):

5^2-9(5)+20
=25-45+20
=0

PRACTICE MAKES PERFECT WITH QUADRATIC EQUATIONS

Solve: x^2+5x+6=0

Step1: FACTOR… Remember that the 2 numbers need to add to 5 and multiply to get 6!

QUICK TIP: Add starts with the letter A, which is the first in the alphabet, so remember that the 2 numbers that need to be added, need to be added to equal the first number! This hopefully will help you in knowing which number is the add and which one is the multiply! 🙂

(x+2)(x+3)=0

Step 2: SOLUTION

x=-2 or x=-3

(alternate formating: x=-2,-3)

Step 3: CHECK

Check for x=-2:

-2^2+5(-2)+6
=4-10+6
=0

Check for x=-3:

-3^2+5(-3)+6
=9+-15+6
=0

Solve: x^2-3=2x

I know what you might be thinking, “what is this??” Don’t worry! We can easily get this equal to zero and in quadratic expression form. All we have to do is move everything over to the left and set equation to zero. Don’t forget when you move a number to another side of the equal sign, the sign changes. Look below:

x^2-2x-3=0

Step 1: FACTOR

(x-3)(x+1)=0

Step 2: SOLUTION

x=3 or x=-1

(alternate formating: x=3,-1)

Step 3: CHECK

Check for x=3:

3^2-2(3)-3
=9-6-3
=0

Check for x=-1:

-1^2-2(-1)-3
=1+2-3
=0

Solve: (x+9)(x+18)=-1

Most of you might think this is already factored, but it’s not! Don’t forget that all quadratics must be equal to ZERO before you can solve by factoring.

Now that you know it has to be set to zero, what might you have to do?

Hopefully, you said to yourself multiply and simplify the left hand side of the equation first and then move the -1 over to the left setting the equation equal to zero. I always simplify using the FOIL method. First, Last, First, Last, Outer, Inner, Inner, Outer. That means, you multiply the x’s, then multiply the 4 and 2, next multiply the 2 and the x, and lastly multiply the 4 and the x. Take a look below:

(x+4)(x+2)=-1

x^2+8+2x+4x=-1

Next, you need to simplify the 2x and 4x, because they both have one x, they can be added together. So your simplified version of the equation will look like this:

x^2+8+6x=-1

Next, move the -1 over and set the equation equal to zero.

x^2+6x+9=0

NOTE: Remember when you move over a number, you switch the signs, so in this case, the -1 became a positive 1 and 1+8 = 9

Now, you can factor and find your solution.

STEP 1: FACTOR

(x+3)(x+3)= 0

STEP 2: SOLUTION

x=-3

STEP 3: CHECK

Check for x=-3

-3^2+6(-3)+9
=9-18+9
=0

Solve: x^2+5x=0

Ahh, it’s a two term quadratic instead of a three term… What to do, what to do??

This is actually pretty easy, so don’t fret! First see if anything can be factored out… and in this case, the x can be factored out! Look below:

STEP 1: FACTOR

x(x+5)=0

You need to remember it’s okay to factor with one variable, but you have to set both factors with x, equal to zero. See below:

x=0, x+5=0

STEP 2: SOLUTION

x=0, -5

STEP 3: CHECK

Check for x=-5

-5^2+5(-5)
=25-25
=0

Now that we have gone through all of the different ways to see a quadratic, you should now know how to factor ANY quadratic equation!

2017-10-13T12:17:57+00:00