The right side of each equation can be expressed as follows.
or
or
or
The result is the quadratic formula,which is abbreviated as follows.
x
b 2 b^2 4 ac
2 a
x
b 2 b^2 4 ac
2 a
x=
- b
2 a
-
2 b^2 - 4 ac
2 a
x=
- b
2 a
+
2 b^2 - 4 ac
2 a
x+
b
2 a
=
- 2 b^2 - 4 ac
2 a
x+
b
2 a
=
2 b^2 - 4 ac
2 a
506 CHAPTER 9 Quadratic Equations, Inequalities, and Functions
If , the
same two solutions
are obtained.a 60Quadratic FormulaThe solutions of the equation are given by
x
b 2 b^2 4 ac
2 a
.
ax^2 +bx+ c= 0 1 with aZ 02
CAUTION In the quadratic formula, the square root is added to or subtracted
from the value of before dividing by 2 a.
OBJECTIVE 2 Solve quadratic equations by using the quadratic formula.
Using the Quadratic Formula (Rational Solutions)Solve
This equation is in standard form, so we identify the values of a, b, and c. Here
a, the coefficient of the second-degree term, is 6, and b, the coefficient of the first-
degree term, is. The constant cis Now substitute into the quadratic formula.
Quadratic formulaSimplify the radical.Take the square root.There are two solutions, one from the sign and one from the sign.
or
Check each solution in the original equation. The solution set is E-^12 ,^43 F.
x=
5 - 11
12
=
- 6
12
=-
1
2
x=
5 + 11
12
=
16
12
=
4
3
+ -
x=
5 11
12
x=
5 2121
12
x=
5 225 + 96
12
x = a=6, b=-5, c=- 4
- 1 - 52 21 - 522 - 41621 - 42
2162
x=
- b 2 b^2 - 4 ac
2 a
- 5 - 4.
6 x^2 - 5 x- 4 =0.
EXAMPLE 1
b
NOW TRYUse parentheses and
substitute carefully
to avoid errors.NOW TRY
EXERCISE 1
Solve. 2 x^2 + 3 x- 20 = 0
NOW TRY ANSWER
- E-4,^52 F