Quadratic Equations: How to Solve by Factoring

Quadratic Equations: How to Solve by Factoring

Last time, I discussed how to add fractions with unlike denominators . Today, I will teach you how to solve quadratic equations by factori...


Last time, I discussed how to add fractions with unlike denominators. Today, I will teach you how to solve quadratic equations by factoring

Why solve by factoring? 


It is easy to solve linear equations with one variable because it only requires basic mathematical operations which are addition, subtraction, multiplication, and division. 

Linear equation examples:

  • 5x + 2 = 4x - 2 
  • x - 1 = 9 - 4x
  • 3 ( x - 2) = 0
But how about quadratic equations where a variable has an exponent of 2 like x+ 7x + 1 = 2x - 5? This is when factoring comes into picture. Later, we will solve this equation by factoring.

Solve by Factoring process has five major steps:

1. Transpose all terms to one side of the equation.
2. Combine like terms.
3. Factor the equation completely.
4. Set each factor to 0.
5. Solve the two resulting equations.

Solve by Factoring (Step by Step) 

Example: x2 + 7x + 1 = 2x - 5

1The first step is to transpose all terms to one side of the equation. When you transpose terms to the other side, reverse the mathematical operation. Therefore, 2x and - 5, when transposed to the other side, becomes - 2x and 5, respectively. Hence, we now have:

x2 + 7x - 2x + 1 + 5 = 0

2The second step is to combine like terms. Like terms are terms which have the same variable and exponent. For example, 7x and -2x are like terms because they have the same variable which is x and same exponent which is 1. This does not apply to coefficients since (1) they don't have a variable beside them and (2) they all have 1 as their exponent. Thefore, we now have:

x2 + 5x + 6 = 0

3The third step is to factor the equation completely. You can follow the format below to solve by factoring easily. 

Factored Form Format:

( x +/- _ ) ( x +/- _ ) = 0

Factored Form Format Explained:

The factored form of a quadratic equation may be Nx + n or Nx - n, where N and n can be any number. If N is 1, we do not write the 1. We only write the variable instead. If you can't see a number beside a variable, then it is presumed to be 1. 

The plus or minus sign in this format means that it can be either plus or minus. Furthermore, the number beside plus or minus sign is missing because we don't know it just yet.

Moving on, we will now solve x2 + 5x + 6 = 0 as indicated in Step 2. In this particular case, think of two numbers that when you add together becomes 5 and when multiplied becomes 6. 

Yes, the two numbers are 2 and 3. Ergo, we get:   

( x +/- 2 ) ( x +/- 3 ) = 0

To decide whether it should be a plus or minus sign, consider 5x and 6. To come up with 5x, 2x and 3x should be added and to come up with 6, 2 and 3 should be multiplied. Hence, we get the plus sign for both equations. 

Not sure if the sign should be plus or minus? Imagine if the other equation is x - 3 = 0. -3x plus 2x is -x and -3 times 2 is -6, when in fact it should be 5x and 6, respectively.

Moving on, we now have: 

( x + 2 ) ( x + 3 ) = 0

4The fourth step is to set each factor to 0. Hence, we get:

x + 2 = 0 and x + 3 = 0

5The fifth and final step is to solve the two resulting equations which gives us:

= - 2 and x = - 3

Therefore, x = - 2, - 3

Want to know if you solve by factoring correctly? Here's how:

Substitute either - 2 or - 3 in x2 + 5x + 6 = 0.

x = -2

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

x = -3

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

If you learned how to solve by factoring after reading this, kindly share this article or leave a comment below. 

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