Solving a Quadratic Equation Using Factoring
(When the Leading Coefficient is Not 1)
(KS4, Year 10)

The Lesson

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.

Lesson Slides

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

Factoring, Factorising

To write a quadratic equation as a product of two brackets is called 'to factor' or 'to factorise' the quadratic equation, depending on country. The method is referred to as 'factoring' or 'factorising'.

Beware

Be Careful with Signs 1

When a quadratic equation has been factored, the roots of the equation can be read off. But remember, you need to flip the sign. For instance, if the factored equation is...
(x + 2)(2x + 3) = 0
...then the roots are:
x = −2 (−ve), x = −32 (−ve)
If the factored equation is...
(x + 2)(2x − 3) = 0
...then the roots are:
x = −2 (−ve), x = 32 (+ve)

Be Careful with Signs 2

Consider the quadratic equation shown below:
ax2 + bx + c
  • b is the coefficient of the x term.
  • c is the constant term.
Depending on the sign of the b and c terms, the numbers you write in the brackets can be positive or negative. The image below defines what is meant by a positive or negative b, c and number in a bracket:

The following table gives a quick summary of what the signs must be:

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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?