1. PROCEDURE FOR SOLVING
EQUATIONS BY FACTORING
Step 1 Make one side zero. Move all nonzero
terms in the equation to one side (say the
left side), so that the other side (right side)
is 0.
Step 2 Factor the left side.
Step 3 Use the zero-product property. Set each
factor in Step 2 equal to 0, and then solve
the resulting equations.
Step 4 Check your solutions.
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2. EXAMPLE 2 Solving an Equation by Factoring
Solve by factoring: 4 2
9x x
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3. EXAMPLE 3 Solving an Equation by Factoring
Solve by factoring: 3 2
2 2x x x   
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4. EXAMPLE 4 Solving a Rational Equation
Solve:
1 1 1
6 1x x
 

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5. EXAMPLE 5
Solving a Rational Equation with an
Extraneous Solution
Solve:
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6. SOLVING EQUATIONS CONTAINING
SQUARE ROOTS
Step 1 Isolate one radical to one side of the
equation.
Step 2 Square both sides of the equation in Step 1
and simplify.
Step 3 If the equation in Step 2 contains a radical,
repeat Steps 1 and 2 to get an equation that
is free of radicals.
Step 5 Check the solutions in the original equation.
Step 4 Solve the equation obtained in Steps 1 - 3.
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7. EXAMPLE 6 Solving Equations Involving Radicals
Solve:
3
6x x x 
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8. EXAMPLE 7 Solving an Equations Involving a Radical
Solve: 2 1 1x x  
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9. EXAMPLE 8 Solving an Equation Involving Two Radicals
Solve: 2 1 1 1x x   
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10. SOLVING EQUATIONS OF THE FORM
um/n = k
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Let m and n be positive integers, k a real number,
and in lowest terms. Then if
m
n

11. EXAMPLE 9 Solving Equations with Rational Exponents
Solve.
Solution
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12. An equation in a variable x is quadratic in
form if it can be written as
EQUATIONS THAT ARE QUADRATIC IN FORM
 2
0 0 ,au bu c a   
where u is an expression in the variable x . We
solve the equation au2 + bu + c = 0 for u.
Then the solutions of the original equation can
be obtained by replacing u by the expression
in x that u represents.
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13. EXAMPLE 10
Solving an Equation That Is Quadratic in
Form
Solve:
Solution
Let u = x1/3, then u2 = (x1/3)2 = x2/3.
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14. EXAMPLE 11
Solving an Equation That Is Quadratic in
Form
Solve:
2
1 1
6 8 0x x
x x
   
       
   
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15. EXAMPLE 12 Investigating Space Travel
Your sister is 5 years older than you are. She
decides she has had enough of Earth and needs
a vacation. She takes a trip to the Omega-One
star system. Her trip to Omega-One and back in
a spacecraft traveling at an average speed v took
15 years, according to the clock and calendar on
the spacecraft. But on landing back on Earth,
she discovers that her voyage took 25 years,
according to the time on Earth.
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16. EXAMPLE 12 Investigating Space Travel
This means that, although you were 5 years
younger than your sister before her vacation,
you are 5 years older than her after her
vacation! Use the time-dilation equation
to calculate the speed of the spacecraft.
2
0 2
1
v
t t
c
 
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17. EXAMPLE 12 Investigating Space Travel
Substitute t0 = 15 (moving-frame time) and
t = 25 (fixed-frame time) to obtain
2
2
2
2
2
2
1 1
3
1
5
9
1
25 5
25
v
c
v
c
v
c
 
 
 
Solution
2
2
2
9
1
25
16
25
v
c
v
c
 
 
 
 
4
5
4
0.8
5
v
c
v c c
 
 
So the spacecraft was moving at
80% (0.8c) the speed of light.
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