Solving Equations Examples And Worksheet Page 9
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14A57
Appendix A.5
Solving Equations
Equations Involving Radicals
Operations such as squaring each side of an equation, raising each side of an equation
to a rational power, and multiplying each side of an equation by a variable quantity all
can introduce extraneous solutions. So, when you use any of these operations, checking
your solutions is crucial.
Example 12
Solving Equations Involving Radicals
a.
2x
7
x
2
Original equation
2x
7
x
2
Isolate radical.
2
2x
7
x
4x
4
Square each side.
2
0
x
2x
3
Write in general form.
0
x
3 x
1
Factor.
x
3
0
x
3
Set 1st factor equal to 0.
x
1
0
x
1
Set 2nd factor equal to 0.
By checking these values, you can determine that the only solution is
x
1.
b.
2x
5
x
3
1
Original equation
2x
5
x
3
1
Isolate
2x
5.
2x
5
x
3
2 x
3
1
Square each side.
2x
5
x
2
2 x
3
Combine like terms.
x
3
2 x
3
Isolate
2 x
3.
When an equation contains two
2
x
6x
9
4 x
3
Square each side.
radicals, it may not be possible
to isolate both. In such cases,
2
x
10x
21
0
Write in general form.
you may have to raise each side
x
3 x
7
0
Factor.
of the equation to a power at two
different stages in the solution,
x
3
0
x
3
Set 1st factor equal to 0.
as shown in Example 12(b).
x
7
0
x
7
Set 2nd factor equal to 0.
The solutions are
x
3
and
x
7.
Check these in the original equation.
Now try Exercise 129.
Example 13
Solving an Equation Involving a Rational Exponent
2 3
x
4
25
Original equation
3
2
x
4
25
Rewrite in radical form.
2
x
4
15,625
Cube each side.
±
x
4
125
Take square root of each side.
x
129,
x
121
Add 4 to each side.
Now try Exercise 137.
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