Solving a Quadratic Equation Using a Difference of Squares
(KS4, Year 10)
The LessonSome 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).
How to Factor a Quadratic Equation Using a Difference of SquaresA 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 SquaresSolving a quadratic equation using a difference of squares is easy.
QuestionSolve the quadratic equation shown below using a difference of squares.
Rewrite the quadratic equation as a difference of squares. In our example, 9 = 3 × 3 = 32.
x2 − 9 = x2 − 32
Compare the difference of squares with the formula to find a.
a = 3
Use the formula to factor the difference of squares:
x2 − a2 = (x + a)(x − a)
Substitute a = 3 into the formula.
x2 − 32 = (x + 3)(x − 3)
Rewrite the quadratic equation.
Equate the first bracket to 0 and solve to find x.
x + 3 = 0 ⇒ x = −3
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 SlidesThe 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 SquaresA 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, FactorisingTo 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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