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

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

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: factor_quadratic_equations_difference_of_squares_formula

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. solve quadratic equation difference of squares example

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. solve quadratic equation difference of squares step 2
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.

solve quadratic equation difference of squares step 5
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.

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: difference_of_squares_numbers In general, we can use symbols instead of numbers: difference_of_squares 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'.

Interactive Widget

You can use this interactive widget to create a graph of your quadratic equation. Use the buttons to change the values of the quadratic equation.
Change the Equation
y
x2
x
y-intercept
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This page was written by Stephen Clarke.

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