In this lesson, students will learn how to factor quadratic expressions by using the difference of two squares.
A quadratic expression is a polynomial that can be written in the form shown below.
A specific type of quadratic expression is called the difference of squares.
Notice that the constant term, 16, is a perfect square, because the square root of 16 is 4, which is an integer. Therefore, you can rewrite your expression as the following:
This is what is called a difference of squares expression, because there are two values squared, x and the integer, with subtraction between the two.
Difference of squares expressions can be factored similarly to other quadratic expressions. Suppose you want to factor the difference of squares expression from the example above:
To factor, you need to find two numbers that multiply to the constant term, and add to the coefficient of the middle term.
Note, when the x term is absent from a quadratic expression, the coefficient of the x term is 0. Therefore, you can write the x term with the coefficient of 0. However, because anything multiplied by 0 is 0, the entire term is 0, so in reality, you don't need to write anything for the x term.
You'll notice that in the expression that you want to factor, your constant term, 16, is a perfect square. Therefore, you can now write your expression as:
To factor your expression, you need to find two numbers that add to 0 and multiply to -16:
Going back to your expression, 4 and -4 are opposites, so they sum to 0, and they also multiply to negative 16. Therefore, you can factor your expression as:
Notice that the integer being squared in the original expression, 4, is part of both factors: x plus 4 in the first factor, and x minus 4 in the second factor,. Therefore, in general, you can factor difference of squares equations as the shown below:
You can verify that you have factored the expression correctly by performing FOIL to see if you get your original expression:
Since you do indeed arrive back at your original expression, you have factored your expression correctly.
Source: This work is adapted from Sophia author Colleen Atakpu.
a^2 -b^2 = (a+b)(a-b)