College Algebra 9th txtbk

SECTION 1–6 Additional Equation-Solving Techniques 97
1
102. CONSTRUCTION A 4-mile track for racing stock cars consists
of two semicircles connected by parallel straightaways (see the figure).
In order to provide sufficient room for pit crews, emergency
vehicles, and spectator parking, the track must enclose an area of
100,000 square feet. Find the length of the straightaways and the diameter
of the semicircles to the nearest foot. [Recall: The area A
and circumference C of a circle of diameter d are given by
A d 2 4 and c d.]
100,000 square feet
1-6 Additional Equation-Solving Techniques
Z Solving Equations Involving Radicals
Z Revisiting Equations Involving Absolute Value
Z Solving Equations of Quadratic Type
In this section, we’ll study equations that are not quadratic but can be transformed into
quadratic equations. We can then solve the quadratic equation, and with a little bit of interpretation,
use the solutions to solve the original equation.
Z Solving Equations Involving Radicals
In solving an equation involving a radical, like
it seems reasonable that we can remove the radical by squaring each side and then proceed
to solve the resulting quadratic equation. Let’s give it a try:
x 2 x 2 0
(x 2)(x 1) 0
or
or
Now we check these results in the original equation.
Check: x 2
x 1x 2
2 ? 12 2
2 ? 14
2 ✓ 2
x 1x 2
x 2 (1x 2) 2
x 2 x 2
x 2 0
x 2
x 1x 2
Square both sides.
Recall that (1a) 2 a if a 0.
Subtract x 2 from both sides.
Factor the left side.
Use the zero product property.
x 1 0
Check: x 1
x 1x 2
1 ? 11 2
1 ? 11
1 1
x 1
That’s interesting: 2 is a solution, but 1 is not. These results are a special case of
Theorem 1.