Solving a Quadratic Equation Using a Difference of Squares
(KS4, Year 10)

The Lesson

Some quadratic equations are in the form of a difference of squares:

The quadratic equation above is a difference of squares because a square number (a number multiplied by itself) is subtracted by another. The two square numbers are:
  • x2 = x × x (x squared).
  • 9 = 32 = 3 × 3 (3 squared).
A quadratic equation in the form of a difference of squares can be factored into two brackets multiplying each other:

How to Factor a Quadratic Equation Using a Difference of Squares

A quadratic equation in the form of a difference of squares can be factored into two brackets:

How to Solve Quadratic Equations Using a Difference of Squares

Solving a quadratic equation using a difference of squares is easy.

Question

Solve the quadratic equation shown below using a difference of squares.

Step-by-Step:

1

Rewrite the quadratic equation as a difference of squares. In our example, 9 = 3 × 3 = 32.
x2 − 9 = x2 − 32

2

Compare the difference of squares with the formula to find a.

a = 3

3

Use the formula to factor the difference of squares:
x2 − a2 = (x + a)(x − a)

4

Substitute a = 3 into the formula.
x2 − 32 = (x + 3)(x − 3)

5

Rewrite the quadratic equation.

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.

6

Equate the first bracket to 0 and solve to find x.
x + 3 = 0 ⇒ x = −3

7

Equate the second bracket to 0 and solve to find x.
x − 3 = 0 ⇒ x = 3

Answer:

We have factored the quadratic equation using a difference of squares: x2 − 9 = (x + 3)(x − 3) = 0. We have solved the quadratic equation: x = −3, x = 3.

Lesson Slides

The slider below shows another real example of how to solve a quadratic equation using a difference of squares. Open the slider in a new tab

What Is a Difference of Squares

A difference of squares is square number (a number multiplied by itself) subtracted from another square number. An example of a difference of squares using numbers is:

In general, we can use symbols instead of numbers:

This can be factored into two brackets:
a2 − b2 = (a + b)(a − b)

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 the country. The method is refered to as 'factoring' or 'factorising'.
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See Also

What is a quadratic equation? What is a square number? What is the FOIL method?