How to Solve a Quadratic Equation Using Factoring (When the Leading Coefficient is Not 1)

Solving a Quadratic Equation Using Factoring (When the Leading Coefficient Is Not 1)

Factoring (or factorising) is a way of simplifying a quadratic equation.

In this lesson, we will look at quadratic equations where the leading coefficient (the number in front of the x2 term) is not 1.

Factoring a quadratic equation writes it as two brackets multiplying each other.

Factoring is the opposite of expanding the two brackets out using the FOIL method.

How to Solve Quadratic Equations Using Factoring (When the Leading Coefficient is Not 1)

Solving a quadratic equation using factoring is easy.

Question

Solve the quadratic equation shown below using factoring.

Step-by-Step:

1

Multiply the coefficent of the x2 term (2) with the constant (6).

2 × 6 = 12

2

Find the pairs of numbers that multiply to make the answer (12).

12 = 1 × 12

12 = 2 × 6

12 = 3 × 4

Don't forget: We have found the factors of 12.

3

Look at the pairs of factors found in Step 2.

Do any of them add up to 7?

7 is the coefficient of the x term in the quadratic equation.

1 + 12 = 13

2 + 6 = 8

3 + 4 = 7

3 and 4 add up to make 7.

4

Rewrite 7x as a sum of two x-terms, using the pairs of factors found in Step 3.

5

Group the terms for factoring.

6

Factor each bracket by taking the greatest common factor out.

7

Each term in brackets should be the same.

If it is not, go back to Step 5 and regroup the terms.

If the terms in the brackets are the same, we can group the terms outside the brackets into their own bracket.

We have factored the quadratic equation.

Check you have factored the quadratic equation correctly by expanding the brackets using the FOIL method and seeing if you get back to the original equation.

Now lets use the factored quadratic equation to solve the quadratic equation.

8

Equate the first bracket to 0 and solve to find x.

x + 2 = 0 ⇒ x = −2

9

Equate the second bracket to 0 and solve to find x.

2x + 3 = 0 ⇒ x = −32

Answer:

We have factored the quadratic equation:

2x2 + 7x + 6 = (x + 2)(2x + 3) = 0.

We have solved the quadratic equation:

x = −2, x = −32.

Slider

The slider below shows another real example of how to solve a quadratic equation using factoring, when the leading coefficient is not 1.

Open the slider in a new tab

See Also

What is the FOIL method? What is a factor? What is a constant? What is a coefficient? Factoring an expression What is the greatest common factor?