Solving Equations Examples And Worksheet Page 5
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Appendix A.5
Solving Equations
Example 4
Solving a Quadratic Equation by Factoring
2
a.
2x
9x
7
3
Original equation
2
2x
9x
4
0
Write in general form.
2x
1 x
4
0
Factor.
1
2x
1
0
x
Set 1st factor equal to 0.
2
x
4
0
x
4
Set 2nd factor equal to 0.
1
The solutions are
x
and
x
4.
Check these in the original equation.
2
2
b.
6x
3x
0
Original equation
3x 2x
1
0
Factor.
3x
0
x
0
Set 1st factor equal to 0.
1
2x
1
0
x
Set 2nd factor equal to 0.
2
1
The solutions are
x
0
and
x
.
Check these in the original equation.
2
Now try Exercise 49.
Note that the method of solution in Example 4 is based on the Zero-Factor Property
from Appendix A.1. Be sure you see that this property works only for equations written
in general form (in which the right side of the equation is zero). So, all terms must
be collected on one side before factoring. For instance, in the equation
x
5 x
2
8,
it is incorrect to set each factor equal to 8. Try to solve this
equation correctly.
Example 5
Extracting Square Roots
Solve each equation by extracting square roots.
2
2
a.
4x
12
b.
x
3
7
Solution
2
a.
4x
12
Write original equation.
2
x
3
Divide each side by 4.
±
x
3
Extract square roots.
When you take the square root of a variable expression, you must account for both
positive and negative solutions. So, the solutions are
x
3
and
x
3.
Check
these in the original equation.
2
b.
x
3
7
Write original equation.
±
x
3
7
Extract square roots.
±
x
3
7
Add 3 to each side.
±
The solutions are
x
3
7.
Check these in the original equation.
Now try Exercise 65.
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