CONFIDENTIAL 1 Grade 8 Algebra1 Solving Quadratic Equations by Factoring.
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Transcript of CONFIDENTIAL 1 Grade 8 Algebra1 Solving Quadratic Equations by Factoring.
![Page 1: CONFIDENTIAL 1 Grade 8 Algebra1 Solving Quadratic Equations by Factoring.](https://reader036.fdocuments.in/reader036/viewer/2022062308/56649dca5503460f94ac0b6b/html5/thumbnails/1.jpg)
CONFIDENTIAL 1
Grade 8 Algebra1Grade 8 Algebra1
Solving Quadratic Solving Quadratic EquationsEquations
by Factoringby Factoring
![Page 2: CONFIDENTIAL 1 Grade 8 Algebra1 Solving Quadratic Equations by Factoring.](https://reader036.fdocuments.in/reader036/viewer/2022062308/56649dca5503460f94ac0b6b/html5/thumbnails/2.jpg)
CONFIDENTIAL 2
Warm UpWarm Up
Solve each equation by graphing the related function.
1) x2 - 49 = 0
2) x2 = x + 12
3) - x2 + 8x = 15
![Page 3: CONFIDENTIAL 1 Grade 8 Algebra1 Solving Quadratic Equations by Factoring.](https://reader036.fdocuments.in/reader036/viewer/2022062308/56649dca5503460f94ac0b6b/html5/thumbnails/3.jpg)
CONFIDENTIAL 3
You have solved quadratic equations bygraphing. Another method used to solvequadratic equations is to factor and use
the Zero Product Property.
Zero Product PropertyZero Product Property
Notice that when writing a quadratic function as its related quadratic equation,
you replace y with 0. So y = 0.
y = ax2 + bx + c
0 = ax2 + bx + c
ax2 + bx + c = 0
![Page 4: CONFIDENTIAL 1 Grade 8 Algebra1 Solving Quadratic Equations by Factoring.](https://reader036.fdocuments.in/reader036/viewer/2022062308/56649dca5503460f94ac0b6b/html5/thumbnails/4.jpg)
CONFIDENTIAL 4
One way to solve a quadratic equation in standard form is to graph the related function and find the x-values where y = 0. In other words, find the zeros of the related function. Recall that a quadratic function
may have two, one, or no zeros.
Using the Zero Product PropertyUsing the Zero Product Property
WORDS NUMBERS ALGEBRA
If the product of two quantities equals zero,
at least one of thequantities equals zero.
3 (0) = 0
0(4) = 0
If ab = 0,
then a = 0 or b = 0.
![Page 5: CONFIDENTIAL 1 Grade 8 Algebra1 Solving Quadratic Equations by Factoring.](https://reader036.fdocuments.in/reader036/viewer/2022062308/56649dca5503460f94ac0b6b/html5/thumbnails/5.jpg)
CONFIDENTIAL 5
Solving Quadratic Equations by GraphingSolving Quadratic Equations by GraphingUse the Zero Product Property to solve each
equation. Check your answer.
A) (x - 3)(x + 7) = 0
x - 3 = 0 or x + 7 = 0
x= 3 or x = -7
Use the Zero Product Property.
Solve each equation.
Check
(x - 3)(x + 7) = 0
(3 - 3)(3 + 7) 0
(0)(10) 0
0 0
(-7 - 3)(x + 7) = 0
(-7 - 3)(-7 + 7) 0
(10)(0) 0
0 0
Substitute each
solution for xinto the original
equation.
The solutions are 3 and -7.
![Page 6: CONFIDENTIAL 1 Grade 8 Algebra1 Solving Quadratic Equations by Factoring.](https://reader036.fdocuments.in/reader036/viewer/2022062308/56649dca5503460f94ac0b6b/html5/thumbnails/6.jpg)
CONFIDENTIAL 6
B) (x)(x - 5) = 0
x = 0 or x - 5 = 0
x= 5
Use the Zero Product Property.
Solve each equation.
Check
(x)(x - 5) = 0
(0)(0 - 5) 0
(0)(-5) 0
0 0
Substitute each
solution for xinto the original
equation.
The solutions are 0 and 5.
(x)(x - 5) = 0
(5)(5 - 5) 0
(5)(0) 0
0 0
![Page 7: CONFIDENTIAL 1 Grade 8 Algebra1 Solving Quadratic Equations by Factoring.](https://reader036.fdocuments.in/reader036/viewer/2022062308/56649dca5503460f94ac0b6b/html5/thumbnails/7.jpg)
CONFIDENTIAL 7
Now you try!
Use the Zero Product Property to solve each equation. Check your answer.
1a. (x)(x + 4) = 0
1b. (x + 4)(x - 3) = 0
![Page 8: CONFIDENTIAL 1 Grade 8 Algebra1 Solving Quadratic Equations by Factoring.](https://reader036.fdocuments.in/reader036/viewer/2022062308/56649dca5503460f94ac0b6b/html5/thumbnails/8.jpg)
CONFIDENTIAL 8
If a quadratic equation is written in standard form, a x 2 + bx + c = 0,
then to solve the equation, you may need to factor before using the
Zero Product Property.
![Page 9: CONFIDENTIAL 1 Grade 8 Algebra1 Solving Quadratic Equations by Factoring.](https://reader036.fdocuments.in/reader036/viewer/2022062308/56649dca5503460f94ac0b6b/html5/thumbnails/9.jpg)
CONFIDENTIAL 9
Solving Quadratic Equations by FactoringSolving Quadratic Equations by FactoringSolve each quadratic equation by factoring.
A) x2 + 7x + 10 = 0
(x + 5) (x + 2) = 0
x + 5 = 0 or x + 2 = 0 Use the Zero Product Property.
Solve each equation.
Check
x2 + 7x + 10 = 0
(-5)2 + 7(-5) + 10 0
25 - 35 + 10 0
0 0
Substitute each
solution for xinto the original
equation.
The solutions are -5 and -2.
x = -5 or x = -2
Factor the trinomial.
x2 + 7x + 10 = 0
(-2)2 + 7(-2) + 10 0
4 - 14 + 10 0
0 0
![Page 10: CONFIDENTIAL 1 Grade 8 Algebra1 Solving Quadratic Equations by Factoring.](https://reader036.fdocuments.in/reader036/viewer/2022062308/56649dca5503460f94ac0b6b/html5/thumbnails/10.jpg)
CONFIDENTIAL 10
B) x2 + 2x = 8
-8 -8
Use the Zero Product Property.
Solve each equation.
The solutions are -4 and 2.
x = -4 or x = 2
Factor the trinomial.
x2 + 2x = 8
x2 + 2x – 8 = 0
The equation must be written in standard form. So subtract 8 from both sides.
(x + 4) (x - 2) = 0
x + 4 = 0 or x - 2 = 0
![Page 11: CONFIDENTIAL 1 Grade 8 Algebra1 Solving Quadratic Equations by Factoring.](https://reader036.fdocuments.in/reader036/viewer/2022062308/56649dca5503460f94ac0b6b/html5/thumbnails/11.jpg)
CONFIDENTIAL 11
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 + 2x - 8 shows two zeros appear to be -
4 and 2, the same as the solutions from factoring.
![Page 12: CONFIDENTIAL 1 Grade 8 Algebra1 Solving Quadratic Equations by Factoring.](https://reader036.fdocuments.in/reader036/viewer/2022062308/56649dca5503460f94ac0b6b/html5/thumbnails/12.jpg)
CONFIDENTIAL 12
C) x2 + 2x + 1 = 0
Use the Zero Product Property.
Solve each equation.
Both factors result in the same solution, so there is one solution, -1.
x = -1 or x = -1
Factor the trinomial.(x + 1) (x + 1) = 0
x + 1 = 0 or x + 1 = 0
![Page 13: CONFIDENTIAL 1 Grade 8 Algebra1 Solving Quadratic Equations by Factoring.](https://reader036.fdocuments.in/reader036/viewer/2022062308/56649dca5503460f94ac0b6b/html5/thumbnails/13.jpg)
CONFIDENTIAL 13
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 + 2x + 1 shows that one zero appears
to be -1, the same as the solution from factoring.
![Page 14: CONFIDENTIAL 1 Grade 8 Algebra1 Solving Quadratic Equations by Factoring.](https://reader036.fdocuments.in/reader036/viewer/2022062308/56649dca5503460f94ac0b6b/html5/thumbnails/14.jpg)
CONFIDENTIAL 14
D) -2x2 = 18 - 12x
-2 (x - 3) (x - 3) = 0
-2 ≠ 0 or x - 3 = 0 Use the Zero Product Property.
Solve each equation.
Check
The only solution is 3.
x = 3
Factor the trinomial.
-2x2 = 18 - 12x
-2(3)2 18 - 12(3)
-18 18 - 36
0 0
Write the equation in standard form.-2x2 + 12x – 18 = 0
-2( x2 - 6x + 9) = 0 Factor out the GCF, -2.
Substitute 3 into the original equation.
![Page 15: CONFIDENTIAL 1 Grade 8 Algebra1 Solving Quadratic Equations by Factoring.](https://reader036.fdocuments.in/reader036/viewer/2022062308/56649dca5503460f94ac0b6b/html5/thumbnails/15.jpg)
CONFIDENTIAL 15
Now you try!
Solve each quadratic equation by factoring. Check your answer.
2a. x2 - 6x + 9 = 0
2b. x2 + 4x = 5
![Page 16: CONFIDENTIAL 1 Grade 8 Algebra1 Solving Quadratic Equations by Factoring.](https://reader036.fdocuments.in/reader036/viewer/2022062308/56649dca5503460f94ac0b6b/html5/thumbnails/16.jpg)
CONFIDENTIAL 16
Sports ApplicationSports ApplicationThe height of a diver above the water during a dive can be modeled by h = -16t2 + 8t + 48, where h is
height in feet and t is time in seconds. Find the time it takes for the diver to reach the water.
h = -16t2 + 8t + 48
0 = -16t2 + 8t + 48
Use the Zero Product Property.
Solve each equation.
Factor the trinomial.
The diver reaches the water when h = 0.
Factor out the GCF, -8.0 = -8(2t2 - t - 6)
0 = -8(2t + 3) (t -2)
-8 ≠ 0, 2t + 3 = 0 or t - 2 = 0
2t = -3 or t = 2
t = -3 2
Since time cannot be negative, (-3/2 ) does not make sense in this situation.
![Page 17: CONFIDENTIAL 1 Grade 8 Algebra1 Solving Quadratic Equations by Factoring.](https://reader036.fdocuments.in/reader036/viewer/2022062308/56649dca5503460f94ac0b6b/html5/thumbnails/17.jpg)
CONFIDENTIAL 17
It takes the diver 2 seconds to reach the water.
Check
0 = -16 t 2 + 8t + 48
0 -16(2)2 + 8(2) + 48
0 -64 + 16 + 48
0 0
Substitute 3 into the original equation.
![Page 18: CONFIDENTIAL 1 Grade 8 Algebra1 Solving Quadratic Equations by Factoring.](https://reader036.fdocuments.in/reader036/viewer/2022062308/56649dca5503460f94ac0b6b/html5/thumbnails/18.jpg)
CONFIDENTIAL 18
Now you try!
3.) 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.
![Page 19: CONFIDENTIAL 1 Grade 8 Algebra1 Solving Quadratic Equations by Factoring.](https://reader036.fdocuments.in/reader036/viewer/2022062308/56649dca5503460f94ac0b6b/html5/thumbnails/19.jpg)
CONFIDENTIAL 19
BREAK
![Page 20: CONFIDENTIAL 1 Grade 8 Algebra1 Solving Quadratic Equations by Factoring.](https://reader036.fdocuments.in/reader036/viewer/2022062308/56649dca5503460f94ac0b6b/html5/thumbnails/20.jpg)
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CONFIDENTIAL 21
Assessment
1) (x + 2) (x - 8) = 0
Use the Zero Product Property to solve each equation. Check your answer.
2) (x - 6) (x - 5) = 0
3) (x + 7) (x + 9) = 0
4) (x) (x - 1) = 0
![Page 22: CONFIDENTIAL 1 Grade 8 Algebra1 Solving Quadratic Equations by Factoring.](https://reader036.fdocuments.in/reader036/viewer/2022062308/56649dca5503460f94ac0b6b/html5/thumbnails/22.jpg)
CONFIDENTIAL 22
Solve each quadratic equation by factoring. Check your answer
6) 3x2 - 4x + 1 = 0
5) 30x = -9x2 - 25
8) x2 - 8x - 9 = 0
7) x2 + 4x - 12 = 0
![Page 23: CONFIDENTIAL 1 Grade 8 Algebra1 Solving Quadratic Equations by Factoring.](https://reader036.fdocuments.in/reader036/viewer/2022062308/56649dca5503460f94ac0b6b/html5/thumbnails/23.jpg)
CONFIDENTIAL 23
9) A group of friends tries to keep a beanbag from touching the ground without using their hands. Once the beanbag has been kicked, its height can be modeled by h = -16t2 + 14t + 2, where h is the height in feet above the ground and t is the time in seconds. Find the time it
takes the beanbag to reach the ground.
![Page 24: CONFIDENTIAL 1 Grade 8 Algebra1 Solving Quadratic Equations by Factoring.](https://reader036.fdocuments.in/reader036/viewer/2022062308/56649dca5503460f94ac0b6b/html5/thumbnails/24.jpg)
CONFIDENTIAL 24
You have solved quadratic equations bygraphing. Another method used to solvequadratic equations is to factor and use
the Zero Product Property.
Zero Product PropertyZero Product Property
Notice that when writing a quadratic function as its related quadratic equation,
you replace y with 0. So y = 0.
y = ax2 + bx + c
0 = ax2 + bx + c
ax2 + bx + c = 0
Let’s review
![Page 25: CONFIDENTIAL 1 Grade 8 Algebra1 Solving Quadratic Equations by Factoring.](https://reader036.fdocuments.in/reader036/viewer/2022062308/56649dca5503460f94ac0b6b/html5/thumbnails/25.jpg)
CONFIDENTIAL 25
One way to solve a quadratic equation in standard form is to graph the related function and find the x-values where y = 0. In other words, find the zeros of the related function. Recall that a quadratic function
may have two, one, or no zeros.
Using the Zero Product PropertyUsing the Zero Product Property
WORDS NUMBERS ALGEBRA
If the product of two quantities equals zero,
at least one of thequantities equals zero.
3 (0) = 0
0(4) = 0
If ab = 0,
then a = 0 or b = 0.
![Page 26: CONFIDENTIAL 1 Grade 8 Algebra1 Solving Quadratic Equations by Factoring.](https://reader036.fdocuments.in/reader036/viewer/2022062308/56649dca5503460f94ac0b6b/html5/thumbnails/26.jpg)
CONFIDENTIAL 26
Solving Quadratic Equations by GraphingSolving Quadratic Equations by GraphingUse the Zero Product Property to solve each
equation. Check your answer.
A) (x - 3)(x + 7) = 0
x - 3 = 0 or x + 7 = 0
x= 3 or x = -7
Use the Zero Product Property.
Solve each equation.
Check
(x - 3)(x + 7) = 0
(3 - 3)(3 + 7) 0
(0)(10) 0
0 0
(-7 - 3)(x + 7) = 0
(-7 - 3)(-7 + 7) 0
(10)(0) 0
0 0
Substitute each
solution for xinto the original
equation.
The solutions are 3 and -7.
![Page 27: CONFIDENTIAL 1 Grade 8 Algebra1 Solving Quadratic Equations by Factoring.](https://reader036.fdocuments.in/reader036/viewer/2022062308/56649dca5503460f94ac0b6b/html5/thumbnails/27.jpg)
CONFIDENTIAL 27
Solving Quadratic Equations by FactoringSolving Quadratic Equations by FactoringSolve each quadratic equation by factoring.
A) x2 + 7x + 10 = 0
(x + 5) (x + 2) = 0
x + 5 = 0 or x + 2 = 0 Use the Zero Product Property.
Solve each equation.
Check
x2 + 7x + 10 = 0
(-5)2 + 7(-5) + 10 0
25 - 35 + 10 0
0 0
Substitute each
solution for xinto the original
equation.
The solutions are -5 and -2.
x = -5 or x = -2
Factor the trinomial.
x2 + 7x + 10 = 0
(-2)2 + 7(-2) + 10 0
4 - 14 + 10 0
0 0
![Page 28: CONFIDENTIAL 1 Grade 8 Algebra1 Solving Quadratic Equations by Factoring.](https://reader036.fdocuments.in/reader036/viewer/2022062308/56649dca5503460f94ac0b6b/html5/thumbnails/28.jpg)
CONFIDENTIAL 28
B) x2 + 2x = 8
-8 -8
Use the Zero Product Property.
Solve each equation.
The solutions are -4 and 2.
x = -4 or x = 2
Factor the trinomial.
x2 + 2x = 8
x2 + 2x – 8 = 0
The equation must be written in standard form. So subtract 8 from both sides.
(x + 4) (x - 2) = 0
x + 4 = 0 or x - 2 = 0
![Page 29: CONFIDENTIAL 1 Grade 8 Algebra1 Solving Quadratic Equations by Factoring.](https://reader036.fdocuments.in/reader036/viewer/2022062308/56649dca5503460f94ac0b6b/html5/thumbnails/29.jpg)
CONFIDENTIAL 29
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 + 2x - 8 shows two zeros appear to be -
4 and 2, the same as the solutions from factoring.
![Page 30: CONFIDENTIAL 1 Grade 8 Algebra1 Solving Quadratic Equations by Factoring.](https://reader036.fdocuments.in/reader036/viewer/2022062308/56649dca5503460f94ac0b6b/html5/thumbnails/30.jpg)
CONFIDENTIAL 30
Sports ApplicationSports ApplicationThe height of a diver above the water during a dive can be modeled by h = -16t2 + 8t + 48, where h is
height in feet and t is time in seconds. Find the time it takes for the diver to reach the water.
h = -16t2 + 8t + 48
0 = -16t2 + 8t + 48
Use the Zero Product Property.
Solve each equation.
Factor the trinomial.
The diver reaches the water when h = 0.
Factor out the GCF, -8.0 = -8(2t2 - t - 6)
0 = -8(2t + 3) (t -2)
-8 ≠ 0, 2t + 3 = 0 or t - 2 = 0
2t = -3 or t = 2
t = -3 2
Since time cannot be negative, (-3/2 ) does not make sense in this situation.
![Page 31: CONFIDENTIAL 1 Grade 8 Algebra1 Solving Quadratic Equations by Factoring.](https://reader036.fdocuments.in/reader036/viewer/2022062308/56649dca5503460f94ac0b6b/html5/thumbnails/31.jpg)
CONFIDENTIAL 31
It takes the diver 2 seconds to reach the water.
Check
0 = -16 t 2 + 8t + 48
0 -16(2)2 + 8(2) + 48
0 -64 + 16 + 48
0 0
Substitute 3 into the original equation.
![Page 32: CONFIDENTIAL 1 Grade 8 Algebra1 Solving Quadratic Equations by Factoring.](https://reader036.fdocuments.in/reader036/viewer/2022062308/56649dca5503460f94ac0b6b/html5/thumbnails/32.jpg)
CONFIDENTIAL 32
You did a great job You did a great job today!today!