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

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.factor quadratic equations a not equal to 1Factoring 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.
solve quadratic equation a not 1 factor example

Step-by-Step:

1

Multiply the coefficent of the x2 term (2) with the constant (6). solve quadratic equation a not 1 factor step 1
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? solve quadratic equation a not 1 factor step 3 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.

solve quadratic equation a not 1 factor step 4

5

Group the terms for factoring.

solve quadratic equation a not 1 factor step 5

6

Factor each bracket by taking the greatest common factor out.

solve quadratic equation a not 1 factor step 6

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.

solve quadratic equation a not 1 factor step 7
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.

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:quadratic_signs_definition_a_not_1The following table gives a quick summary of what the signs must be:

quadratic signs table
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This page was written by Stephen Clarke.