# confidential 1 solving quadratic equations by factoring solving quadratic equations by factoring

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CONFIDENTIAL 1

Solving Quadratic Solving Quadratic EquationsEquations

by Factoringby Factoring

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

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

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.

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.

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

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

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.

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

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

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.

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

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.

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.

CONFIDENTIAL 15

Now you try!

Solve each quadratic equation by factoring. Check your answer.

2a. x2 - 6x + 9 = 0

2b. x2 + 4x = 5

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.

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.

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.

CONFIDENTIAL 19

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

CONFIDENTIAL 20

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

CONFIDENTIAL 21

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.

CONFIDENTIAL 22

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

CONFIDENTIAL 23

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.

CONFIDENTIAL 24

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.

CONFIDENTIAL 25

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

CONFIDENTIAL 26

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

CONFIDENTIAL 27

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.

CONFIDENTIAL 28

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.

CONFIDENTIAL 29

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.

CONFIDENTIAL 30

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