Homework: Part I

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Homework 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
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Homework Part II
5. 2x2 12x 14 0
6. x2 18x 81 0
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.
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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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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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Additional 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
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Additional Example 1A Continued
Use the Zero Product Property to solve the
equation. Check your answer.
Substitute each solution for x in the original
equation.
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Additional 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
Substitute each solution for x in the original
equation.
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Partner Share! 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
Substitute each solution for x in the original
equation.
?
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Partner Share! Example 1b
Use the Zero Product Property to solve the
equation. Check your answer.
(x 4)(x 3) 0
x 4 0 or x 3 0
Use the Zero Product Property.
x 4 or x 3
Solve each equation.
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Partner Share! 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 in the original
equation.
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You may need to factor before using the Zero
Product Property. You can check your answers by
substituting into the original equation or by
graphing. If the factored form of the equation
has two different factors, the graph of the
related function will cross the x-axis in two
places. If the factored form has two identical
factors, the graph will cross the x-axis in one
place.
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Additional 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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Additional 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.
x 7 or x 3
Solve each equation.
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Additional Example 2B Continued
Check Graph the related quadratic function.
Because there are two solutions found by
factoring, the graph should cross the x-axis in
two places.
The graph of y x2 4x 21 intersects the
x-axis at x 7 and x 3, the same as the
solutions from factoring. ?
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Additional 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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Additional Example 2C Continued
Check Graph the related quadratic function.
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Additional 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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Additional 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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Partner Share! Example 2a
Solve the quadratic equation by factoring. Check
your answer.
x2 6x 9 0
Factor the trinomial.
(x 3)(x 3) 0
Use the Zero Product Property.
x 3 0 or x 3 0
x 3 or x 3
Solve each equation.
The only solution is 3.
Substitute 3 into the original equation.
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Partner Share! 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.
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Partner Share! Example 2b Continued
Check Graph the related quadratic function.
Because there are two solutions found by
factoring, the graph should cross the x-axis in
two places.
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Partner Share! 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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Partner Share! Example 2c Continued
Check Graph the related quadratic function.
Because there is one solution found by factoring,
the graph should cross the x-axis in one place.
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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
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Additional 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 GCF, 8.
0 8(2t 1)(t 1)
Factor the trinomial.
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Additional 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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Partner Share! 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 GCF, 8.
0 8(2t 3)(t 1)
Factor the trinomial.
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Partner Share! 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 1.5 into the original equation.
?
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Lesson Review 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 Review Part II
1, 7
5. 2x2 12x 14 0
9
6. x2 18x 81 0
2
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 seconds