Algebra Demystified 2nd Ed

Chapter 10 QuadratiC EQuations 373

10. 80 16 32 0
    1
    16

80 16 32

2

2

xx

xx

−−=

()−−=^1

16

(0)

5

() () ()()

()

xx^2

x

−−=

=−− ±−−−=

20

1145 2

25

(^211140)
10
141
10
±−−()= ±

11. 18 39 20 0
    39 39 41820
    218

39 1521

2

2

xx

x

++=

=−± − ()()=−± −

()

11440

36

39 81

36

39 9

36

39 9

36

39 9

36

30

36

 ,=−± =−±=−+−−=− ,,

 ,

−

=−−

48

36

5

6

4

3

12. xx

x

2

2

10 25 0

10 10 4125

21

10 100 1

++=

=−± − ()()=−± −

()

000

2

10 0

2

10

2

5

=−±

=− =−

Rational Equations That Lead to Quadratic Equations

Some rational equations (an equation with one or more fractions as terms)

become quadratic equations once we multiply each term by the least common

denominator (LCD). Remember, we must be sure that any solutions do not lead

to a zero in a denominator in the original equation.

Generally, we use one of two main approaches to clear the denominator(s) in

a rational equation. If the equation is in the form of “fraction = fraction,” cross-

multiply. If the equation is not in this form, we multiply each side of the equation

by LCD. Finding the LCD often means factoring each denominator completely.

We learned in Chapter 7 to multiply both sides of an equation by the LCD and

then to distribute the LCD. In this chapter, we will simply multiply each term by

the LCD, eliminating one step. Next, we collect all of the terms on one side of the

equation, leaving a 0 on the other side. In the examples and practice problems

below, the solutions that lead to a zero in a denominator will be stated.