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:
- x^{2} = x × x (x squared).
- 9 = 3^{2} = 3 × 3 (3 squared).
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 = 3^{2}.
x^{2} − 9 = x^{2} − 3^{2}
2
Compare the difference of squares with the formula to find a.
a = 3
3
Use the formula to factor the difference of squares:
x^{2} − a^{2} = (x + a)(x − a)
4
Substitute a = 3 into the formula.
x^{2} − 3^{2} = (x + 3)(x − 3)
5
Rewrite 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: x^{2} − 9 = (x + 3)(x − 3) = 0. We have solved the quadratic equation: x = −3, x = 3.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:
a^{2} − b^{2} = (a + b)(a − b)