How to Solve Quadratic Equations using Factoring Method

This is the easiest method of solving a quadratic equation as long as the binomial or trinomial is easily factorable. Otherwise, we will need other methods such as completing the square or using the quadratic formula.

The following diagram illustrates the main approach to solving a quadratic equation by factoring method.


Main Idea of using Factoring Method to Solve a Quadratic Equation

If I have an equation with one side containing factors of a given polynomial and is equal to zero, then I can solve for the variable by setting each factor equal to zero then solve.

The diagram above suggests the following key points:

  • One side of the equation is just zero.
  • The opposite side should contain the factors of the given polynomial.
  • After the two conditions stated above are met, then it is now OKAY to set each factor equal to zero then solve for the value of the unknown variable.

Examples of How to Solve Quadratic Equations using the Factoring Method

Example 1: Solve the quadratic equation below by Factoring Method.

the quantity x plus 7 times the quantity x minus 2 equals 0

I consider this type of problem as a “freebie” because it is already set up for us to find the solutions. Notice that the left side contains factors of some polynomial, and the right side is just zero!

What we need to do is simply set each factor equal to zero, and solve each equation for x.

x is equal to negative 7 and x is equal to positive 2

The answers are x = - \,7 and x = 2. You may back-substitute these values of x to the original equation to verify if they are true answers. I will leave it to you as an exercise.


Example 2: Solve the quadratic equation below by Factoring Method.

negative 2 x squared plus 6x is equal to 0

The left side of the equation is a binomial. That means I can pull out a monomial factor. If you think about it, between the numerical coefficients - \,2 and 6, I can factor out - \,2. More so, between {x^2} and x, I can factor out x. So to find the overall factor (it’s like finding the GCF), I will multiply - \,2 and x to get - \,2x.

Note, I can also factor out 2x instead of - \,2x. The final answer should be the same. Try it out!

x is equal to 0 and x is equal to 3

Example 3: Solve the quadratic equation below using the Factoring Method.

x squared plus 3x minus 10 is equal to 0

Have you factored a trinomial before where the coefficient of the squared term is + 1? If not, it is very simple.

To factor this trinomial into two binomials, I need to find two numbers (by trial and error) that satisfy two given conditions:

  • The product of these two numbers is equal to the constant term (last number) which is - 10.
  • The sum of these two numbers is equal to the coefficient of the linear term which is + 3.
Multiply two numbers to get negative 10 and add two numbers to get positive 3.

Since the product of two numbers is negative, I know that these numbers must have opposite signs. More so, having a sum of positive number implies that the number with the larger absolute value must be positive.

If you work it out mentally or using paper and pencil to run through possible combinations, the two numbers that can satisfy the given conditions are + 5 and - 2.

To check, their products \left( { + \,5} \right)\left( { - \,2} \right) = - 10, and their sum \left( { + \,5} \right)\,\, + \,\,\left( { - \,2} \right) = + 3. Works out great!

x is equal to negative 5 and x is equal to 2

The final solutions are x = - \,5 and x = 2.


Example 4: Solve the quadratic equation below using the Factoring Method.

3 x cubed minus 27x is equal to 0

Between the coefficients 3 and - 27, I can pull out 3. And between {x^3} and x, I can take out x. Therefore the overall expression that I can factor out is their product: \left( 3 \right)\left( x \right) = 3x.

Notice that after I factored out 3x, I am left with a “special” binomial called the “Difference of Two Squares” which is very easy to factor.

x squared minus 9 is equal to x squared minus 3 squared

It is always the case that the middle signs will be opposites (see yellow).

Here is the complete solution.

x is equal to 0 or x is equal to negative 3 or x is equal to 3

You should back-substitute to verify that x = 0, x = - \,3, and x = 3 are the correct solutions.


Example 5: Solve the quadratic equation below using the Factoring Method.

x squared plus 5x plus 6 is equal to 20

The first thing I realize in this problem is that one side of the equation doesn’t contain zero. I can easily create a zero on the right side by subtracting both sides by 20.

After doing so, the left side should have a factorable trinomial that is very similar to problem 3.

To factor out this trinomial, think of two numbers when multiplied together gives - 14 (constant term) and when added gives + 5 (coefficient of x-term). By trial and error, the numbers should be - 2 and 7. You may verify this correct combination.

x is equal to 2 and x is equal to negative 7

The final answers are x = 2 and x = - \,7.


Example 6: Solve the quadratic equation below using the Factoring Method.

x squared minus x minus 42 is equal to 0

Solution:

x is equal to negative 6 and x is equal to 7

Here we have x = - \,6 and x = 7 as our final answers.


Example 7: Solve the quadratic equation below using the Factoring Method.

x squared minus 6x plus 4 is equal to negative 1

Solution:

x is equal to 5 and x is equal to 1

Our final answers are x = 5 and x = 1.


Example 8: Solve the quadratic equation below using the Factoring Method.

2x squared plus 4x minus 6 is equal to 0

Solution:

x is equal to 1 and x is equal to negative 3

The final solutions are x = 1 and x = - \,3.


Practice with Worksheets


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Solving Quadratic Equations by Square Root Method
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Solving Quadratic Equations by Completing the Square