Solving Quadratic Equations

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Solving Quadratic Equations by Factoring
9-6
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 1
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• Warm Up
• Find each product.
• 1. (x 2)(x 7) 2. (x 11)(x 5)
• 3. (x 10)2
• Factor each polynomial.
• 4. x2 12x 35 5. x2 2x 63
• 6. x2 10x 16 7. 2x2 16x 32

x2 9x 14
x2 6x 55
x2 20x 100
(x 5)(x 7)
(x 7)(x 9)
(x 2)(x 8)
2(x 4)2
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Objective
Solve quadratic equations by factoring.
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You have solved quadratic equations by graphing.
Another method used to solve quadratic equations
is to factor and use the Zero Product Property.
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Example 1A Use the Zero Product Property
Use the Zero Product Property to solve the
equation. Check your answer.
(x 7)(x 2) 0
Use the Zero Product Property.
x 7 0 or x 2 0
Solve each equation.
x 7 or x 2
The solutions are 7 and 2.
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Example 1A Continued
Use the Zero Product Property to solve the
equation. Check your answer.
Substitute each solution for x into the original
equation.
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Example 1B Use the Zero Product Property
Use the Zero Product Property to solve each
equation. Check your answer.
(x 2)(x) 0
Use the Zero Product Property.
(x)(x 2) 0
x 0 or x 2 0
Solve the second equation.
x 2
The solutions are 0 and 2.
Substitute each solution for x into the original
equation.
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Check It Out! Example 1a
Use the Zero Product Property to solve each
equation. Check your answer.
(x)(x 4) 0
Use the Zero Product Property.
x 0 or x 4 0
Solve the second equation.
x 4
The solutions are 0 and 4.
Substitute each solution for x into the original
equation.
?
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Check It Out! Example 1b
Use the Zero Product Property to solve the
equation. Check your answer.
(x 4)(x 3) 0
Use the Zero Product Property.
x 4 0 or x 3 0
x 4 or x 3
Solve each equation.
The solutions are 4 and 3.
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Check It Out! Example 1b Continued
Use the Zero Product Property to solve the
equation. Check your answer.
(x 4)(x 3) 0
Substitute each solution for x into the original
equation.
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If a quadratic equation is written in standard
form, ax2 bx c 0, then to solve the
equation, you may need to factor before using the
Zero Product Property.
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Example 2A Solving Quadratic Equations by
Factoring
Solve the quadratic equation by factoring. Check
your answer.
x2 6x 8 0
(x 4)(x 2) 0
Factor the trinomial.
x 4 0 or x 2 0
Use the Zero Product Property.
Solve each equation.
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Example 2B Solving Quadratic Equations by
Factoring
Solve the quadratic equation by factoring. Check
your answer.
x2 4x 21
The equation must be written in standard form. So
subtract 21 from both sides.
(x 7)(x 3) 0
Factor the trinomial.
x 7 0 or x 3 0
Use the Zero Product Property.
Solve each equation.
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Example 2B Continued
Solve the quadratic equation by factoring. Check
your answer.
x2 4x 21
Check Graph the related quadratic function. The
zeros of the related function should be the same
as the solutions from factoring.
The graph of y x2 4x 21 shows that two
zeros appear to be 7 and 3, the same as the
solutions from factoring. ?
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Example 2C Solving Quadratic Equations by
Factoring
Solve the quadratic equation by factoring. Check
your answer.
x2 12x 36 0
(x 6)(x 6) 0
Factor the trinomial.
x 6 0 or x 6 0
Use the Zero Product Property.
x 6 or x 6
Solve each equation.
Both factors result in the same solution, so
there is one solution, 6.
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Example 2C Continued
Solve the quadratic equation by factoring. Check
your answer.
x2 12x 36 0
Check Graph the related quadratic function.
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Example 2D Solving Quadratic Equations by
Factoring
Solve the quadratic equation by factoring. Check
your answer.
2x2 20x 50
The equation must be written in standard form. So
add 2x2 to both sides.
2x2 20x 50 0
Factor out the GCF 2.
2(x2 10x 25) 0
Factor the trinomial.
2(x 5)(x 5) 0
2 ? 0 or x 5 0
Use the Zero Product Property.
x 5
Solve the equation.
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Example 2D Continued
Solve the quadratic equation by factoring. Check
your answer.
2x2 20x 50
Check
Substitute 5 into the original equation.
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Check It Out! Example 2a
Solve the quadratic equation by factoring. Check
your answer.
x2 6x 9 0
Factor the trinomial.
(x 3)(x 3) 0
x 3 0 or x 3 0
Use the Zero Product Property.
x 3 or x 3
Solve each equation.
Both equations result in the same solution, so
there is one solution, 3.
Substitute 3 into the original equation.
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Check It Out! Example 2b
Solve the quadratic equation by factoring. Check
your answer.
x2 4x 5
Write the equation in standard form. Add 5 to
both sides.
(x 1)(x 5) 0
Factor the trinomial.
x 1 0 or x 5 0
Use the Zero Product Property.
x 1 or x 5
Solve each equation.
The solutions are 1 and 5.
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Check It Out! Example 2b Continued
Solve the quadratic equation by factoring. Check
your answer.
x2 4x 5
Check Graph the related quadratic function. The
zeros of the related function should be the same
as the solutions from factoring.
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Check It Out! Example 2c
Solve the quadratic equation by factoring. Check
your answer.
30x 9x2 25
Write the equation in standard form.
9x2 30x 25 0
1(9x2 30x 25) 0
Factor out the GCF, 1.
1(3x 5)(3x 5) 0
Factor the trinomial.
1 ? 0 or 3x 5 0
Use the Zero Product Property. 1 cannot equal 0.
Solve the remaining equation.
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Check It Out! Example 2c Continued
Solve the quadratic equation by factoring. Check
your answer.
30x 9x2 25
Check Graph the related quadratic function. The
zeros of the related function should be the same
as the solutions from factoring.
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Check It Out! Example 2d
Solve the quadratic equation by factoring. Check
your answer.
3x2 4x 1 0
(3x 1)(x 1) 0
Factor the trinomial.
3x 1 0 or x 1 0
Use the Zero Product Property.
Solve each equation.
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Check It Out! Example 2d Continued
Solve the quadratic equation by factoring. Check
your answer.
3x2 4x 1 0
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Example 3 Application
The height in feet of a diver above the water can
be modeled by h(t) 16t2 8t 8, where t is
time in seconds after the diver jumps off a
platform. Find the time it takes for the diver to
reach the water.
h 16t2 8t 8
The diver reaches the water when h 0.
0 16t2 8t 8
0 8(2t2 t 1)
Factor out the GFC, 8.
0 8(2t 1)(t 1)
Factor the trinomial.
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Example 3 Continued
Use the Zero Product Property.
8 ? 0, 2t 1 0 or t 1 0
2t 1 or t 1
Solve each equation.
It takes the diver 1 second to reach the water.
Check 0 16t2 8t 8
Substitute 1 into the original equation.
?
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Check It Out! Example 3
What if? The equation for the height above the
water for another diver can be modeled by h
16t2 8t 24. Find the time it takes this
diver to reach the water.
h 16t2 8t 24
The diver reaches the water when h 0.
0 16t2 8t 24
0 8(2t2 t 3)
Factor out the GFC, 8.
0 8(2t 3)(t 1)
Factor the trinomial.
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Check It Out! Example 3 Continued
Use the Zero Product Property.
8 ? 0, 2t 3 0 or t 1 0
?
2t 3 or t 1
Solve each equation.
Since time cannot be negative, 1 does not make
sense in this situation.
t 1.5
It takes the diver 1.5 seconds to reach the water.
Check 0 16t2 8t 24
Substitute 1 into the original equation.
?
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Lesson Quiz Part I

• Use the Zero Product Property to solve each
  equation. Check your answers.
• 1. (x 10)(x 5) 0
• 2. (x 5)(x) 0
• Solve each quadratic equation by factoring. Check
  your answer.
• 3. x2 16x 48 0
• 4. x2 11x 24

10, 5
5, 0
4, 12
3, 8
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Lesson Quiz Part II
1, 7
5. 2x2 12x 14 0
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6. x2 18x 81 0
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7. 4x2 16x 16
8. The height of a rocket launched upward from a
160 foot cliff is modeled by the function h(t)
16t2 48t 160, where h is height in feet and
t is time in seconds. Find the time it takes the
rocket to reach the ground at the bottom of the
cliff.
5 s