5.4 The Quadratic Formula - College of the Redwoods

486 Chapter 5 Quadratic Functions
It’s interesting to note that this problem could have been solved by factoring. Indeed,
x 2 + 6x − 27 = 0
(x − 3)(x + 9) = 0,
so the zero product property requires that either x − 3 = 0 or x + 9 = 0, which leads
to x = 3 or x = −9, answers identical to those found by the quadratic formula.
We’ll have more to say about the “discriminant” soon, but it’s no coincidence that
the quadratic x 2 + 6x − 27 factored. Here is the relevant fact.
When the Discriminant is a Perfect Square. In the quadratic formula,
x = −b ± √ b 2 − 4ac
,
2a
the number under the radical, b 2 − 4ac, is called the discriminant. When the
discriminant is a perfect square, the quadratic function will always factor.
However, it is not always the case that we can factor the given quadratic. Let’s look
at another example.
◮ Example 13.
of f(x) = 2.
Given the quadratic function f(x) = x 2 − 2x, find all real solutions
Because f(x) = x 2 − 2x, the equation f(x) = 2 becomes
x 2 − 2x = 2.
Set one side of the equation equal to zero by subtracting 2 from both sides of the
equation. 8 x 2 − 2x − 2 = 0
Compare x 2 − 2x − 2 = 0 with the general quadratic equation ax 2 + bx + c = 0 and
note that a = 1, b = −2 and c = −2. Write down the quadratic formula.
x = −b ± √ b 2 − 4ac
2a
Next, substitute a = 1, b = −2, and c = −2. Note the careful use of parentheses. 9
x = −(−2) ± √ (−2) 2 − 4(1)(−2)
2(1)
8 Note that the quadratic expression on the left-hand side of the resulting equation does not factor over
the integers. There are no integer pairs whose product is −2 that sum to −2.
9 For example, without parentheses, −2 2 = −4, whereas with parentheses (−2) 2 = 4.
Version: Fall 2007