College Algebra 9th txtbk

90 CHAPTER 1 EQUATIONS AND INEQUALITIES
The result is known as the quadratic formula:
Z THEOREM 1 Quadratic Formula
If ax 2 bx c 0, a 0, then
x b 2b2 4ac
2a
The quadratic formula should be memorized and used to solve quadratic equations when
other methods fail, or are more difficult to apply.
EXAMPLE 5 Using the Quadratic Formula
Solve 2x 3 2 x 2 using the quadratic formula. Leave the answer in simplest radical form.
2x 3 2 x 2
SOLUTION Multiply both sides by 2.
4x 3 2x 2
Write in standard form.
2x 2 4x 3 0
x b 2b2 4ac
2a
(4) 2(4)2 4(2)(3)
2(2)
4 140
4
4 2110
4
Identify a, b, and c and use the quadratic
formula: a 2, b 4, c 3
2 110
2
ZZZ CAUTION ZZZ 1. 4 2 (4) 2
4 2 16 and (4) 2 16
2.
3.
2 110
2
4 2110
4
2 110
2
2110
2 110
2 4 110
2
4 2110
4
2(2 110)
4
2 110
2
MATCHED PROBLEM 5 Solve x 2 5 2 3x by use of the quadratic formula. Leave the answer in simplest radical
form.
The expression under the square root in the quadratic formula, b 2 4ac, is called the discriminant.
It gives us useful information about the corresponding roots, as shown in Table 1.
Table 1 Discriminant and Roots
Discriminant Roots of ax 2 bx c 0
b 2 4ac a, b, and c real numbers, a 0
Positive
Two distinct real roots
0 One real root (a double root)
Negative
Two imaginary roots, one the conjugate of the other