Holt McDougal Algebra 1 8-6 Solving Quadratic Equations by Factoring 8-6 Solving Quadratic Equations...
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![Page 1: Holt McDougal Algebra 1 8-6 Solving Quadratic Equations by Factoring 8-6 Solving Quadratic Equations by Factoring Holt Algebra 1 Warm Up Warm Up Lesson.](https://reader030.fdocuments.net/reader030/viewer/2022033006/56649f315503460f94c4d1a1/html5/thumbnails/1.jpg)
Holt McDougal Algebra 1
8-6 Solving Quadratic Equations by Factoring8-6 Solving Quadratic Equations
by Factoring
Holt Algebra 1
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
Holt McDougal Algebra 1
![Page 2: Holt McDougal Algebra 1 8-6 Solving Quadratic Equations by Factoring 8-6 Solving Quadratic Equations by Factoring Holt Algebra 1 Warm Up Warm Up Lesson.](https://reader030.fdocuments.net/reader030/viewer/2022033006/56649f315503460f94c4d1a1/html5/thumbnails/2.jpg)
Holt McDougal Algebra 1
8-6 Solving Quadratic Equations by Factoring
Warm UpFind 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
![Page 3: Holt McDougal Algebra 1 8-6 Solving Quadratic Equations by Factoring 8-6 Solving Quadratic Equations by Factoring Holt Algebra 1 Warm Up Warm Up Lesson.](https://reader030.fdocuments.net/reader030/viewer/2022033006/56649f315503460f94c4d1a1/html5/thumbnails/3.jpg)
Holt McDougal Algebra 1
8-6 Solving Quadratic Equations by Factoring
Solve quadratic equations by factoring.
Objective
![Page 4: Holt McDougal Algebra 1 8-6 Solving Quadratic Equations by Factoring 8-6 Solving Quadratic Equations by Factoring Holt Algebra 1 Warm Up Warm Up Lesson.](https://reader030.fdocuments.net/reader030/viewer/2022033006/56649f315503460f94c4d1a1/html5/thumbnails/4.jpg)
Holt McDougal Algebra 1
8-6 Solving Quadratic Equations by Factoring
You have solved quadratic equations by graphing. Another method used to solve quadratic equations is to factor and use the Zero Product Property.
![Page 5: Holt McDougal Algebra 1 8-6 Solving Quadratic Equations by Factoring 8-6 Solving Quadratic Equations by Factoring Holt Algebra 1 Warm Up Warm Up Lesson.](https://reader030.fdocuments.net/reader030/viewer/2022033006/56649f315503460f94c4d1a1/html5/thumbnails/5.jpg)
Holt McDougal Algebra 1
8-6 Solving Quadratic Equations by Factoring
Example 1A: Use the Zero Product Property
Use the Zero Product Property to solve the equation. Check your answer.
(x – 7)(x + 2) = 0
x – 7 = 0 or x + 2 = 0
x = 7 or x = –2
The solutions are 7 and –2.
Use the Zero Product Property.
Solve each equation.
![Page 6: Holt McDougal Algebra 1 8-6 Solving Quadratic Equations by Factoring 8-6 Solving Quadratic Equations by Factoring Holt Algebra 1 Warm Up Warm Up Lesson.](https://reader030.fdocuments.net/reader030/viewer/2022033006/56649f315503460f94c4d1a1/html5/thumbnails/6.jpg)
Holt McDougal Algebra 1
8-6 Solving Quadratic Equations by Factoring
Example 1A Continued
Use the Zero Product Property to solve the equation. Check your answer.
Substitute each solution for x into the original equation.
Check (x – 7)(x + 2) = 0
(7 – 7)(7 + 2) 0(0)(9) 0
0 0Check (x – 7)(x + 2) = 0
(–2 – 7)(–2 + 2) 0(–9)(0) 0
0 0
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Holt McDougal Algebra 1
8-6 Solving Quadratic Equations by Factoring
Example 1B: Use the Zero Product Property
Use the Zero Product Property to solve each equation. Check your answer.
(x – 2)(x) = 0
x = 0 or x – 2 = 0x = 2
The solutions are 0 and 2.
Use the Zero Product Property.
Solve the second equation.
Substitute each solution for x into
the original equation.
Check (x – 2)(x) = 0
(0 – 2)(0) 0
(–2)(0) 00 0
(x – 2)(x) = 0
(2 – 2)(2) 0 (0)(2) 0
0 0
(x)(x – 2) = 0
![Page 8: Holt McDougal Algebra 1 8-6 Solving Quadratic Equations by Factoring 8-6 Solving Quadratic Equations by Factoring Holt Algebra 1 Warm Up Warm Up Lesson.](https://reader030.fdocuments.net/reader030/viewer/2022033006/56649f315503460f94c4d1a1/html5/thumbnails/8.jpg)
Holt McDougal Algebra 1
8-6 Solving Quadratic Equations by Factoring
Use the Zero Product Property to solve each equation. Check your answer.
Check It Out! Example 1a
(x)(x + 4) = 0
x = 0 or x + 4 = 0x = –4
The solutions are 0 and –4.
Use the Zero Product Property.
Solve the second equation.
Substitute each solution for x into
the original equation.
Check (x)(x + 4) = 0
(0)(0 + 4) 0
(0)(4) 0 0 0
(x)(x +4) = 0
(–4)(–4 + 4) 0(–4)(0) 0
0 0
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Holt McDougal Algebra 1
8-6 Solving Quadratic Equations by Factoring
Check It Out! 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
x = –4 or x = 3
The solutions are –4 and 3.
Use the Zero Product Property.
Solve each equation.
![Page 10: Holt McDougal Algebra 1 8-6 Solving Quadratic Equations by Factoring 8-6 Solving Quadratic Equations by Factoring Holt Algebra 1 Warm Up Warm Up Lesson.](https://reader030.fdocuments.net/reader030/viewer/2022033006/56649f315503460f94c4d1a1/html5/thumbnails/10.jpg)
Holt McDougal Algebra 1
8-6 Solving Quadratic Equations by Factoring
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.
Check (x + 4)(x – 3 ) = 0
(–4 + 4)(–4 –3) 0
(0)(–7) 00 0
Check (x + 4)(x – 3 ) = 0
(3 + 4)(3 –3) 0
(7)(0) 00 0
![Page 11: Holt McDougal Algebra 1 8-6 Solving Quadratic Equations by Factoring 8-6 Solving Quadratic Equations by Factoring Holt Algebra 1 Warm Up Warm Up Lesson.](https://reader030.fdocuments.net/reader030/viewer/2022033006/56649f315503460f94c4d1a1/html5/thumbnails/11.jpg)
Holt McDougal Algebra 1
8-6 Solving Quadratic Equations by Factoring
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.
![Page 12: Holt McDougal Algebra 1 8-6 Solving Quadratic Equations by Factoring 8-6 Solving Quadratic Equations by Factoring Holt Algebra 1 Warm Up Warm Up Lesson.](https://reader030.fdocuments.net/reader030/viewer/2022033006/56649f315503460f94c4d1a1/html5/thumbnails/12.jpg)
Holt McDougal Algebra 1
8-6 Solving Quadratic Equations by Factoring
To review factoring techniques, see lessons 8-3 through 8-5.
Helpful Hint
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Holt McDougal Algebra 1
8-6 Solving Quadratic Equations by Factoring
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
x – 4 = 0 or x – 2 = 0
x = 4 or x = 2The solutions are 4 and 2.
Factor the trinomial.
Use the Zero Product Property.
Solve each equation.
x2 – 6x + 8 = 0
(4)2 – 6(4) + 8 016 – 24 + 8 0
0 0
Checkx2 – 6x + 8 = 0
(2)2 – 6(2) + 8 0 4 – 12 + 8 0
0 0
Check
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Holt McDougal Algebra 1
8-6 Solving Quadratic Equations by Factoring
Example 2B: Solving Quadratic Equations by Factoring
Solve the quadratic equation by factoring. Check your answer.
x2 + 4x = 21x2 + 4x = 21
–21 –21x2 + 4x – 21 = 0
(x + 7)(x –3) = 0
x + 7 = 0 or x – 3 = 0
x = –7 or x = 3
The solutions are –7 and 3.
The equation must be written in standard form. So subtract 21 from both sides.
Factor the trinomial.
Use the Zero Product Property.Solve each equation.
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Holt McDougal Algebra 1
8-6 Solving Quadratic Equations by Factoring
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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Holt McDougal Algebra 1
8-6 Solving Quadratic Equations by Factoring
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
x – 6 = 0 or x – 6 = 0
x = 6 or x = 6
Both factors result in the same solution, so there is one solution, 6.
Factor the trinomial.
Use the Zero Product Property.
Solve each equation.
![Page 17: Holt McDougal Algebra 1 8-6 Solving Quadratic Equations by Factoring 8-6 Solving Quadratic Equations by Factoring Holt Algebra 1 Warm Up Warm Up Lesson.](https://reader030.fdocuments.net/reader030/viewer/2022033006/56649f315503460f94c4d1a1/html5/thumbnails/17.jpg)
Holt McDougal Algebra 1
8-6 Solving Quadratic Equations by Factoring
Example 2C Continued
Solve the quadratic equation by factoring. Check your answer.
x2 – 12x + 36 = 0
Check Graph the related quadratic function.
The graph of y = x2 – 12x + 36 shows that one zero appears to be 6, the same as the solution from factoring.
●
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Holt McDougal Algebra 1
8-6 Solving Quadratic Equations by Factoring
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.
Factor out the GCF 2.
+2x2 +2x2
0 = 2x2 + 20x + 50
–2x2 = 20x + 50
2x2 + 20x + 50 = 0
2(x2 + 10x + 25) = 0
Factor the trinomial.2(x + 5)(x + 5) = 0
2 ≠ 0 or x + 5 = 0
x = –5
Use the Zero Product Property.
Solve the equation.
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Holt McDougal Algebra 1
8-6 Solving Quadratic Equations by Factoring
Example 2D Continued
Solve the quadratic equation by factoring. Check your answer.
–2x2 = 20x + 50
Check
–2x2 = 20x + 50
–2(–5)2 20(–5) + 50–50 –100 + 50–50 –50
Substitute –5 into the original equation.
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Holt McDougal Algebra 1
8-6 Solving Quadratic Equations by Factoring
(x – 3)(x – 3) is a perfect square. Since both factors are the same, you solve only one of them.
Helpful Hint
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Holt McDougal Algebra 1
8-6 Solving Quadratic Equations by Factoring
Check It Out! Example 2a Solve the quadratic equation by factoring. Check your answer.
x2 – 6x + 9 = 0
(x – 3)(x – 3) = 0
x – 3 = 0 or x – 3 = 0
x = 3 or x = 3Both equations result in the same solution, so there is one solution, 3.
Factor the trinomial.
Use the Zero Product Property.
Solve each equation.
x2 – 6x + 9 = 0
(3)2 – 6(3) + 9 09 – 18 + 9 0
0 0
Check
Substitute 3 into the original equation.
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Holt McDougal Algebra 1
8-6 Solving Quadratic Equations by Factoring
Check It Out! Example 2b
Solve the quadratic equation by factoring. Check your answer.
x2 + 4x = 5
x2 + 4x = 5–5 –5
x2 + 4x – 5 = 0
Write the equation in standard form. Add – 5 to both sides.
Factor the trinomial.
Use the Zero Product Property.
Solve each equation.
(x – 1)(x + 5) = 0
x – 1 = 0 or x + 5 = 0
x = 1 or x = –5
The solutions are 1 and –5.
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Holt McDougal Algebra 1
8-6 Solving Quadratic Equations by Factoring
Check It Out! Example 2b Continued
Solve the quadratic equation by factoring. Check your answer.
x2 + 4x = 5Check 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 – 5 shows that the two zeros appear to be 1 and –5, the same as the solutions from factoring.
●●
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Holt McDougal Algebra 1
8-6 Solving Quadratic Equations by Factoring
Check It Out! Example 2c
Solve the quadratic equation by factoring. Check your answer.
30x = –9x2 – 25
–9x2 – 30x – 25 = 0
–1(3x + 5)(3x + 5) = 0
–1(9x2 + 30x + 25) = 0
–1 ≠ 0 or 3x + 5 = 0
Write the equation in standard form.
Factor the trinomial.
Use the Zero Product Property. – 1 cannot equal 0.
Solve the remaining equation.
Factor out the GCF, –1.
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Holt McDougal Algebra 1
8-6 Solving Quadratic Equations by Factoring
Check It Out! Example 2c Continued
Solve the quadratic equation by factoring. Check your answer.
30x = –9x2 – 25Check Graph the related quadratic function. The zeros of the related function should be the same as the solutions from factoring.
The graph of y = –9x2 – 30x – 25 shows one zero and it appears to be at , the same as the solutions from factoring.
●
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Holt McDougal Algebra 1
8-6 Solving Quadratic Equations by Factoring
Check It Out! Example 2d
Solve the quadratic equation by factoring. Check your answer.
3x2 – 4x + 1 = 0
(3x – 1)(x – 1) = 0
or x = 1
Factor the trinomial.
Use the Zero Product Property.
Solve each equation.
3x – 1 = 0 or x – 1 = 0
The solutions are and x = 1.
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Holt McDougal Algebra 1
8-6 Solving Quadratic Equations by Factoring
Check It Out! Example 2d Continued
Solve the quadratic equation by factoring. Check your answer.
3x2 – 4x + 1 = 0
3x2 – 4x + 1 = 0
3(1)2 – 4(1) + 1 0 3 – 4 + 1 0
0 0
Check3x2 – 4x + 1 = 0
3 – 4 + 1 0
0 0
Check
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Holt McDougal Algebra 1
8-6 Solving Quadratic Equations by Factoring
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
0 = –16t2 + 8t + 8
0 = –8(2t2 – t – 1)
0 = –8(2t + 1)(t – 1)
The diver reaches the water when h = 0.
Factor out the GFC, –8.
Factor the trinomial.
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Holt McDougal Algebra 1
8-6 Solving Quadratic Equations by Factoring
Example 3 Continued
–8 ≠ 0, 2t + 1 = 0 or t – 1= 0 Use the Zero Product
Property.
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.
0 –16(1)2 + 8(1) + 8
0 –16 + 8 + 8 0 0
Since time cannot be negative, does not make sense in this situation.
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Holt McDougal Algebra 1
8-6 Solving Quadratic Equations by Factoring
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
0 = –16t2 + 8t + 24
0 = –8(2t2 – t – 3)
0 = –8(2t – 3)(t + 1)
The diver reaches the water when h = 0.
Factor out the GFC, –8.
Factor the trinomial.
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Holt McDougal Algebra 1
8-6 Solving Quadratic Equations by Factoring
–8 ≠ 0, 2t – 3 = 0 or t + 1= 0 Use the Zero Product Property.
2t = 3 or t = –1 Solve each equation.
Since time cannot be negative, –1 does not make sense in this situation.
It takes the diver 1.5 seconds to reach the water.
Check 0 = –16t2 + 8t + 24
Substitute 1 into the original equation.
0 –16(1.5)2 + 8(1.5) + 24
0 –36 + 12 + 24 0 0
Check It Out! Example 3 Continued
t = 1.5
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Holt McDougal Algebra 1
8-6 Solving Quadratic Equations by Factoring
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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Holt McDougal Algebra 1
8-6 Solving Quadratic Equations by Factoring
Lesson Quiz: Part II
1, –7
–9
–2
5 s
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.