Beginning Algebra, 11th Edition

Using the Squaring Property of Equality

Solve

Use the squaring property to square each side.

On the left,.

Subtract 1.

CHECK Original equation

Let

Add.

3 = 3 ✓ True

29  3

28 + 1  3 x=8.

2 x+ 1 = 3

x= 8

x + 1 = (^9) A 2 aB^2 =a
A 2 x+ 1 B
(^2) = 32
2 x+ 1 = 3
2 x+ 1 =3.
EXAMPLE 1
SECTION 8.6 Solving Equations with Radicals 531
OBJECTIVES
Solving Equations with Radicals
8.6
1 Solve radical
equations having
square root radicals.
2 Identify equations
with no solutions.
3 Solve equations
by squaring a
binomial.
4 Solve radical
equations having
cube root radicals.
A radical equationis an equation having a variable in the radicand.
2 x+ 1 = 3 and 32 x= 28 x+ 9 Radical equations
Squaring Property of Equality
If each side of a given equation is squared, then all solutions of the original equa-
tion are amongthe solutions of the squared equation.
CAUTION Using the squaring property can give a new equation with moresolu-
tions than the original equation. For example, starting with and squaring each
side gives
,or
This last equation, has either of twosolutions, 4 or , while the original
equation, has only onesolution, 4.
Because of this possibility, checking is more than just a guard against algebraic
errors when solving an equation with radicals. It is an essential part of the solution
process. All proposed solutions from the squared equation must be checked in the
original equation.
x=4,
x^2 =16, - 4
x^2 = 42 x^2 =16.
x= 4
A check is
essential.
Because this statement is true, is the solution set of Here, the
equation obtained by squaring had just one solution, which also satisfied the original
equation. NOW TRY
586 2 x+ 1 =3.
NOW TRY
EXERCISE 1
Solve 2 x- 5 =6.
NOW TRY ANSWER

1. 5416

OBJECTIVE 1 Solve radical equations having square root radicals.To

solve radical equations having square root radicals, we need a new property, called

the squaring property of equality.