Brain Busters Solve for x. Assume that a and b represent positive real numbers.Evaluate for the given values of a, b, and c. See Section 1.3.
- 101.a=6, b=7, c= 2 102.a=1, b=-6, c= 9
a=3, b=1, c=- 1 a=4, b=11, c=- 32 b^2 - 4 acPREVIEW EXERCISES
9 x^2 - 25 a= 0 15 x- 2 b 22 = 3 a x^2 - a^2 - 36 = 0x^2 - b= 0 x^2 = 4 b 4 x^2 =b^2 + 16SECTION 9.2 The Quadratic Formula 505
OBJECTIVES
The Quadratic Formula
9.2
1 Derive the
quadratic formula.
2 Solve quadratic
equations by using
the quadratic
formula.
3 Use the discriminant
to determine the
number and type
of solutions.
In this section, we complete the square to solve the general quadratic equation
where a, b, and care complex numbers and The solution of this general equa-
tion gives a formula for finding the solution of anyspecific quadratic equation.
OBJECTIVE 1 Derive the quadratic formula.To solve by
completing the square (assuming a 70 ), we follow the steps given in Section 9.1.
ax^2 + bx+ c= 0
aZ0.
ax^2 + bx+c= 0,
Divide by a. (Step 1)Subtract (Step 2)(Step 3)Add fractions.Square root propertyx+ (Step 6)
b
2 a
=
B
b^2 - 4 ac
4 a^2
or x+
b
2 a
=-
B
b^2 - 4 ac
4 a^2
ax+
b
2 a
b
2=
b^2 - 4 ac
4 a^2
ax+
b
2 a
b
2=
b^2
4 a^2
+
- 4 ac
4 a^2
ax+
b
2 a
b
2=
b^2
4 a^2
+
- c
a
x^2 +
b
a
x+
b^2
4 a^2
=-
c
a
+
b^2
4 a^2
c
1
2
a
b
a
bd
2= a
b
2 a
b
2=
b^2
4 a^2
cx a.
(^2) + b
a
x=-
c
a
x^2 +
b
a
x+
c
a
= 0
ax^2 +bx+ c= 0
We can simplify
B
b^2 - 4 ac
4 a^2
as
2 b^2 - 4 ac
24 a^2
, or
2 b^2 - 4 ac
2 a
.
Add b to each side. (Step 4)2
4 a^2Write the left side as a perfect square.
Rearrange the right side. (Step 5)Write with a common denominator.