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10.1 Solving Quadratic Equations
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Solve quadratic equations by factoring.
Solve quadratic equations by the Square Root Property.
Use substitution to solve equations of quadratic form.
What You Will Learn
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Solving Quadratic Equations by
Factoring
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Solving Quadratic Equations by Factoring
The first step in solving a quadratic equation by factoring
is
to write the equation in general form.
Next, factor the left side.
Finally, set each factor equal to zero and solve for x.
Be sure to check each solution in the original equation.
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a. x2 + 5x = 24 Original equation
x2 + 5x – 24 = 0 Write in general form.
(x + 8)(x – 3) = 0 Factor.
x + 8 = 0 x = – 8 Set 1st factor equal to 0.
x – 3 = 0 x = 3 Set 2nd factor equal to 0.
Example 1 – Solving Quadratic Equations by Factoring
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Example 1 – Solving Quadratic Equations by Factoring
b. 3x2 = 4 – 11x Original equation
3x2 + 11x – 4 = 0 Write in general form.
(3x – 1)(x + 4) = 0 Factor.
3x – 1 = 0 x = Set 1st factor equal to 0.
x + 4 = 0 x = – 4 Set 2nd factor equal to 0.
cont’d
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c. 9x2 + 12 = 3 + 12x + 5x2 Original equation
4x2 – 12x + 9 = 0 Write in general form.
(2x – 3)(2x – 3) = 0 Factor.
2x – 3 = 0 x = Set factor equal to 0.
Check each solution in its original equation.
Example 1 – Solving Quadratic Equations by Factoring cont’d
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The Square Root Property
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The Square Root Property
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Example 2 – Using the Square Root Property
a. 3x2 = 15 Original equation
x2 = 5 Divide each side by 3.
Square Root Property
The solutions are and . Check these in
the original equation.
b. (x – 2)2 = 10 Original equation
Square Root Property
Add 2 to each side.
The solutions are and
Check these in the original equation.
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Example 2 – Using the Square Root Property
c. (3x – 6)2 – 8 = 0
(3x – 6)2 = 8
The solutions are and
Check these in the original equation.
Original equation
Add 8 to each side.
Add 6 to each side.
Divide each side by 3.
cont’d
Square Root Property and
rewrite as
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The Square Root Property
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Example 3 – Square Root Property (Complex Square Root)
a. x2 + 8 = 0
x2 = –8
The solutions are and . Check
these in the original equation.
Original equation
Subtract 8 from each side.
Square Root Property
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Example 3 – Square Root Property (Complex Square Root)
b. (x – 4)2 = –3
The solutions are and . Check
these in the original equation.
Original equation
Square Root Property
Add 4 to each side.
cont’d
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Example 3 – Square Root Property (Complex Square Root)
c. 2(3x – 5)2 + 32 = 0
2(3x – 5)2 = –32
(3x – 5)2 = –16
3x – 5 = ±4i
3x = 5 ± 4i
The solutions are and Check these
in the original equation.
Original equation
Subtract 32 from each side.
Divide each side by 2.
Square Root Property
Add 5 to each side.
Divide each side by 3.
cont’d
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Page 502
#’s 1 – 41 every other odd
Homework:
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Equations of Quadratic Form
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Equations of Quadratic Form
Both the factoring method and the Square Root Property
can be applied to nonquadratic equations that are of
quadratic form. An equation is said to be of quadratic form
if it has the form
au2 + bu + c = 0
where u is an algebraic expression.
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Equations of Quadratic Form
Here are some examples.
To solve an equation of quadratic form, it helps to make a
substitution and rewrite the equation in terms of u.
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Example 5 – Solving an Equation of Quadratic Form
Solve x4 – 13x2 + 36 = 0.
Solution
Begin by writing the original equation in quadratic form, as
follows.
x4 – 13x2 + 36 = 0
(x2)2 – 13(x2) + 36 = 0
Next, let u = x2 and substitute u into the equation written
in
quadratic form. Then, factor and solve the equation.
u2 – 13u + 36 = 0
Write original equation.
Write in quadratic form.
Substitute u for x2.
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Example 5 – Solving an Equation of Quadratic Form
(u – 4)(u – 9) = 0
u – 4 = 0 u = 4
u – 9 = 0 u = 9
At this point you have found the “u-solutions.” To find the
“x-solutions,” replace u with x2 and solve for x.
u = 4 x2 = 4 x = ±2
u = 9 x2 = 9 x = ±3
The solutions are x = 2, x = –2, x = 3, and x = –3. Check
these in the original equation.
Factor.
Set 1st factor equal to 0.
Set 2nd factor equal to 0.
cont’d
