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2013-08-30

Quadratic Equations

A.CED.2, A.REI.4, A.APR.3

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Table of Contents

Identify Quadratic EquationsExplain Characteristics of Quadratic EquationsGraphing Quadratic EquationsTransforming Quadratic EquationsSolve Quadratic Equations by GraphingSolve Quadratic Equations by FactoringSolve Quadratic Equations Using Square RootsSolve Quadratic Equations by Completing the Square

Solve Quadratic Equations by Using the Quadratic Formula

Key Terms

Solving Application Problems

The Discriminant

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Key Terms

Return to Table of Contents

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Axis of symmetry: The vertical line that divides a parabola into two symmetrical halves.

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Maximum: The y-value of the vertex if a < 0 and the parabola opens downward

Minimum: The y-value of the vertex if a > 0 and the parabola opens upward

Parabola: The curve result of graphing a quadratic equation

(+ a)Max

Min(- a)

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Quadratic Equation: An equation that can be written in the standard form ax2 + bx + c = 0. Where a, b and c are real numbers and "a" does not = 0.

Vertex: The highest or lowest point on a parabola.

Zero of a Function: An x value that makes the y value equal to zero.

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Characteristics of Quadratic

EquationsReturn to Table of Contents

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A quadratic equation is an equation of the form ax2 + bx + c = 0 , where "a" is not equal to 0.

The form ax 2 + bx + c = 0 is called the standard form of the quadratic equation.

The standard form is not unique. For example, x2 - x + 1 = 0 can be written as the equivalent equation -x2 + x - 1 = 0.

Also, 4x2 - 2x + 2 = 0 can be written as the equivalent equation 2x2 - x + 1 = 0. Why is this equivalent?

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Practice writing quadratic equations in standard form:(Reduce if possible.)Write 2x2 = x + 4 in standard form:

2x2 - x - 4 = 0

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1 Write 3x = -x2 + 7 in standard form:

A

B

C

A. x2 + 3x-7= 0

B. x2 -3x +7=0

C. -x2 -3x -7= 0

Ans

wer

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2 Write 6x2 - 6x= 12 in standard form:

A

B

C

A. 6x2 - 6x -12= 0

B. x2 -x -2=0

C. -x2 + x +2= 0

Ans

wer

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3 Write 3x - 2= 5x in standard form:

A

B

C

a. 2x + 2 = 0

b. -2x -2=0

c. not a quadratic equation

Ans

wer

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The graph of a quadratic is a parabola, a u-shaped figure.

The parabola will open upward or downward.

← upward→downward

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A parabola that opens upward contains a vertex that is a minimum point. A parabola that opens downward contains a vertex that is a maximum point.

vertex

vertex

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The domain of a quadratic function is all real numbers.

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To determine the range of a quadratic function, ask yourself two questions: Is the vertex a minimum or maximum? What is the y-value of the vertex?

If the vertex is a minimum, then the range is all real numbers greater than or equal to the y-value.

The range of this quadratic is -6 to

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If the vertex is a maximum, then the range is all real numbers less than or equal to the y-value.

The range of this quadratic is to 10

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An axis of symmetry (also known as a line of symmetry) will divide the parabola into mirror images. The line of symmetry is always a vertical line. It is found using the following equation:

x=2

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To find the axis of symmetry simply plug the values of a and b into the equation: Remember the form ax 2 + bx + c. In this example a = 2, b = -8 and c =2

x=2

a b c

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The x-intercepts are the points at which a parabola intersects the x-axis. These points are also known as zeroes, roots or solutions and solution sets. Each quadratic equation will have two, one or no real x-intercepts.

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4 The vertical line that divides a parabola into two symmetrical halves is called...

A discriminant

B perfect square

C axis of symmetry

D vertex

E slice

Ans

wer

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5 What are the vertex and axis of symmetry of the parabola shown in the diagram below?

A vertex: (1,−4); axis of symmetry: x = 1

B vertex: (1,−4); axis of symmetry: x = −4

C vertex: (−4,1); axis of symmetry: x = 1

D vertex: (−4,1); axis of symmetry: x = −4

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

Ans

wer

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6 The equation y = x2 + 3x − 18 is graphed on the set of axes below.

AB

C

D

Based on this graph, what are the roots of the equation x2 + 3x − 18 = 0?

−3 and 6

0 and −18

3 and −6

3 and −18


Ans

wer

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7 The equation y = − x 2 − 2x + 8 is graphed on the set of axes below.

A 8 and 0

B 2 and -4

C 9 and -1

D 4 and -2

Based on this graph, what are the roots of the equation− x2 − 2x + 8 = 0?

Ans

wer


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8 What is an equation of the axis of symmetry of the parabola represented by y = −x2 + 6x − 4?

A x = 3

B y = 3

C x = 6

D y = 6


Ans

wer

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9 The height, y, of a ball tossed into the air can be reresented by the equation y = -x2 + 10x +3, where x is the elapsed time. What is the equation of the axis of symmetry of this parabola?

A y = 5

B y = -5

C x = 5

D x = -5


Ans

wer

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10 What is the equation of the axis of symmetry of the parabola shown in the diagram below?

A x = -0.5

B x = 2

C x = 4.5

D x = 13


18 16 14 12 10

8642

1 2 3 4

5

5 6 7 8 9 10

Ans

wer

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Transforming Quadratic Equations


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The quadratic parent equation is y = x 2 . The graph of all other quadratic equations are transformations of the graph of y= x 2 .

x x2

-3 9

-2 4

-1 1

0 0

1 1

2 4

3 9

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The quadratic parent equation is y = x 2. How is y = x2 changed into y = 2x2?

x 2

-3 18

-2 8

-1 2

0 0

1 2

2 8

3 18

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x 0.5-3 4.5

-2 2

-1 0.5

0 0

1 0.5

2 2

3 4.5

The quadratic parent equation is y = x 2. How is y = x2 changed into y = .5x2?

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How does a>0 affect the parabola? How does a<0 affect the parabola?

What does "a" do in y = ax2+ bx + c ?

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What does "a" also do in y =ax2 + bx +c ? How does your conclusion about "a" change as a changes?

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If the absolute value of a is > 1, then the graph of the equation is narrower than the graph of the parent equation.

If the absolute value of a is < 1, then the graph of the equation is wider than the graph of the parent equation.

If a > 0, the graph opens up.

If a < 0, the graph opens down.

What does "a" do in y = ax2+ bx + c ?

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11 Without graphing determine which direction does the parabola open and if the graph is wider or narrower than the parent equation.

A up, wider

B up, narrower

C down, wider

D down, narrower

Ans

wer

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12 Without graphing determine which direction does the parabola open and if the graph is wider or narrower than the parent function.

A up, wider

B up, narrower

C down, wider

D down, narrower

Ans

wer

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13 Without graphing determine which direction does the parabola open and if the graph is wider or narrower than the parent equation.

A up, wider

B up, narrower

C down, wider

D down, narrower

Ans

wer

y = -2x2 + 100x + 45

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A up, wider

B up, narrower

C down, wider

D down, narrower

Ans

wer

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A up, wider

B up, narrower

C down, wider

D down, narrower

Ans

wer

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What does "c" do in y = ax2+ bx + c ?

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"c" moves the graph up or down the same value as "c".

"c" is the y- intercept.

What does "c" do in y = ax2+ bx + c ?

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16 Without graphing, what is the y- intercept of the the given parabola?

Ans

wer

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Ans

wer

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Ans

wer

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Ans

wer

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20 Choose all that apply to the following quadratic:

A opens up

B opens down Ans

wer

C wider than parent function

D narrower than parent function

E y-intercept of y = -4

F y-intercept of y = -2

G y-intercept of y = 0

H y-intercept of y = 2

I y-intercept of y = 4

J y-intercept of y = 6

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21 Choose all that apply to the following quadratic:

A opens up

B opens down

C wider than parent function

D narrower than parent function

Ans

werE y-intercept of y = -4

F y-intercept of y = -2

G y-intercept of y = 0

H y-intercept of y = 2

I y-intercept of y = 4

J y-intercept of y = 6

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22 The diagram below shows the graph of y = −x2 − c.

A B C D

Which diagram shows the graph of y = x2 − c?


Ans

wer

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Graphing Quadratic Equations


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Graph by Following Six Steps:

Step 1 - Find Axis of Symmetry

Step 2 - Find Vertex

Step 3 - Find Y intercept

Step 4 - Find two more points

Step 5 - Partially graph

Step 6 - Reflect

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Step 1 - Find the Axis of Symmetry

Axis of Symmetry

Pull

What is theAxis of Symmetry?

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Graph y = 3x 2 – 6x + 1

The axis of symmetryis x = 1.

a = 3b = -6

Formula:

Step 1 - Find the Axis of Symmetry

x = - (- 6) = 6 = 1 2(3) 6

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a = 3, b = -6 and c = 1

Step 2 - Find the vertex by substituting the value of x (the axis of symmetry) into the equation to get y.

y = 3(1)2 + -6(1) + 1y = 3 - 6 + 1y = -2Vertex = (1 , -2)

y = 3x 2 - 6x + 1

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Step 3 - Find y intercept.

y- intercept

What is they-intercept?

Pull

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The y-intercept is 1 and the graph passes through(0,1).

y = 3x 2 – 6x + 1

The y- intercept is always the c value, because x = 0.

Graph y = 3x 2 – 6x + 1

c = 1y = ax 2 + bx + c

Step 3 - Find y- intercept.

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Choose different values of x and plug in to find points.

Step 4 - Find two more points on the parabola.

Graph y = 3x 2 – 6x + 1 y = 3x 2 – 6x + 1

y = 3(-1) 2 – 6(-1) + 1

y = 3 +6 + 1y = 10

(-1,10)Let's pick x = -1 and x = -2

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y = 3x 2 - 6x + 1

y = 3(-2) 2 - 6(-2) + 1

y = 3(4) + 12 + 1y = 25

(-2,25)

Graph y = 3x 2 – 6x + 1

Step 4 - Find two points (Continued )

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Step 5 - Graph the axis of symmetry, the vertex, the point containing they-intercept and two other points.

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Step 6 - Reflect the points across the axis of symmetry. Connect the points with a smooth curve.

(4,25)

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23 What is the axis of symmetry for y = x 2 + 2x - 3 (Step 1)?

A 1

B -1

Ans

wer

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24 What is the vertex for y = x 2 + 2x - 3(Step 2)?

A (-1,-4)

B (1,-4)

C (-1,4)

Ans

wer

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25 What is the y-intercept for y = x 2 + 2x - 3 (Step 3)?

A -3

B 3

Ans

wer

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Graph y= x2 + 2x -3

axis of symmetry = -1

vertex = -1,-4

y intercept = -3

2 other points (step 4)

(1,0)

(2,5)

Partially graph (step 5)

Reflect (step 6)

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Graph

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Graph

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Graph

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Solve Quadratic Equations by

GraphingReturn to Table of Contents

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One way to solve a quadratic equation in standard form is find the zeros by graphing.

A zero is the point at which the parabola intersects the x-axis.

A quadratic may have one, two or no zeros.

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How many zeros do the parabolas have? What are the values of the zeros?

No zeroes(doesn't cross the "x" axis)

2 zeroes; x = -1 and x=3

1 zero;x=1

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To solve a quadratic equation by graphing follow the 6 step process we already learned.

Step 1 - Find Axis of Symmetry

Step 2 - Find Vertex

Step 3 - Find Y intercept

Step 4 - Find two more points

Step 5 - Partially graph

Step 6 - Reflect

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26

A

B

C y = -2x 2 + 12x - 18

Solve the equation by graphing.

-12x + 18 = -2x 2

Which of these is in standard form?

y = 2x2 - 12x + 18

y = -2x 2 - 12x + 18

Ans

wer

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27 What is the axis of symmetry?

Ans

wer

y = -2x 2 + 12x - 18

A -3

B 3

C 4

D -5

-b2a

Formula

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28

What is the vertex?

y = -2x 2 + 12x - 18

A (3,0)

B (-3,0)

C (4,0)

D (-5,0)

Ans

wer

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29

What is the y- intercept?

y = -2x 2 + 12x - 18

A (0,0)

B (0, 18)

C (0, -18)

D (0, 12)

Ans

wer

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30

A B

C D

If two other points are (5,-8) and (4,-2), what does the graph of y = -2x2 + 12x - 18 look like?

Ans

wer

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31

What is(are) the zero(s)?

y = -2x 2 + 12x - 18

A -18

B 4

C 3

D -8

Ans

wer

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Solve Quadratic Equations by

FactoringReturn to Table of Contents

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Solving Quadratic Equationsby Factoring

Review of factoring - To factor a quadratic trinomial of the form x2 + bx + c, find two factors of c whose sum is b.

Example - To factor x 2 + 9x + 18, look for factors whose sum is 9.Factors of 18 Sum

1 and 18 19

2 and 9 11

3 and 6 9 x 2 + 9x + 18 = (x + 3)(x + 6)

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When c is positive, its factors have the same sign.

The sign of b tells you whether the factors are positive or negative.

When b is positive, the factors are positive.When b is negative, the factors are negative.

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Remember the FOIL method for multiplying binomials

1. Multiply the F irst terms (x + 3)(x + 2) x x = x 2

2. Multiply the O uter terms (x + 3)(x + 2) x 2 = 2x 3. Multiply the Inner terms (x + 3)(x + 2) 3 x = 3x

4. Multiply the Last terms (x + 3)(x + 2) 3 2 = 6

(x + 3)(x + 2) = x 2 + 2x + 3x + 6 = x 2 + 5x + 6

F O I L

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Zero Product Property

For all real numbers a and b, if the product of two quantities equals zero, at least one of the quantities equals zero.

Numbers Algebra

3(0) = 0 If ab = 0,

4(0) = 0 Then a = 0 or b = 0

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Example 1: Solve x2 + 4x - 12 = 0

(x + 6) (x - 2) = 0

x + 6 = 0 or x - 2 = 0 - 6 - 6 + 2 +2 x = -6 x = 2

-62 + 4(-6) - 12 = 0 -62 + (-24) - 12 = 0 36 - 24 - 12 = 0 0 = 0 or22 + 4(2) - 12 = 0 4 + 8 - 12 = 0 0 = 0

Use "FUSE" !

Factor the trinomial using the FOIL method.

E qual - problem solved! The solutions are -6 and 2.

U se the Zero property

S ubstitue found value into original equation

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Example 2: Solve x2 + 36 = 12x -12x -12x The equation has to be

written in standard form (ax2 + bx + c). So subtract 12x from both sides.

x 2 - 12x + 36 = 0

(x - 6)(x - 6) = 0

x - 6 = 0 +6 +6 x = 6

6 2 + 36 = 12(6)

36 + 36 = 72

72 = 72


E qual - problem solved!



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Example 3: Solve x2 - 16x + 48= 0

(x - 4)(x -12) = 0

x - 4 = 0 x -12 = 0 +4 +4 +12 +12 x = 4 x = 12

4 2 - 16(4) + 48 = 0 16 - 64 + 48 = 0 -48+48 = 0 0 = 0

12 2 - 16(12) + 48 = 0144 -192 + 48 = 0 -48 + 48 = 0 0 = 0

-48


E qual - problem solved!



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32 Solve

A -7

B -5

C -3

D -2

E 2

F 3

G 5

H 6

I 7

J 15

Ans

wer

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33 Solve

A -7

B -5

C -3

D -2

E 2

F 3

G 5

H 6

I 7

J 15

Ans

wer

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34 Solve

A -12

B -4

C -3D -2

E 2

F 3

G 4

H 6

I 8J 12

Ans

wer

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35 Solve

A -7

B -5C -3D 35E 12

F 0

G 5

H 6

I 7J 37

Ans

wer

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36 Solve

A -3/4

B -1/2

C -4/3

D -2

E 2

F 3/4

G 1/2

H 4/3

I -3

J 3

Ans

wer

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37 Which equation has roots of −3 and 5?

A x2 + 2x − 15 = 0

B x2 − 2x − 15 = 0

C x2 + 2x + 15 = 0

D x2 − 2x + 15 = 0


Ans

wer

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Solve Quadratic Equations Using

Square Roots


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You can solve a quadratic equation by the square root method if you can write it in the form: x² = c

If x and c are algebraic expressions, then: x = or x =

written as: x = ±

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Solve for z: z² = 49z = ± z = ±7

The solution set is 7 and -7

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A quadratic equation of the form x2 = c can be solved using the Square Root Property.

Example: Solve 4x2 = 20

4x 2 = 20 4 4 x 2 = 5

The solution set is and -

Divide both sides by 4 to isolate x²

x = ±

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Solve 5x² = 20 using the square root method:

5x2 = 20 5 5

x2 = 4

x = or x = - x = ±2

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Solve (2x - 1)² = 20 using the square root method.

or

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38 When you take the square root of a real number, your answer will always be positive.

True

False Ans

wer

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39 If x2 = 16, then x =

A 4

B 2

C -2

D 26

E -4

Ans

wer

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40 If y2 = 4, then y =

A 4

B 2

C -2

D 26

E -4

Ans

wer

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41 If 8j2 = 96, then j =

A

B

C

D

E

Ans

wer

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42 If 4h2 -10= 30, then h =

A

B

C

D

E

Ans

wer

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43 If (3g - 9)2 + 7= 43, then g =

A

B

C

D

E

Ans

wer

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Solving Quadratic Equations by

Completing the Square


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Before we can solve the quadratic equation, we first have to find the missing value of C. To do this, simply take the value of b, divide it in 2 and then square the result.

Find the value that completes the square.

100

64

1

(b/2)2

8/2 = 4

42 = 16

ax2+bx+c

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44 Find (b/2)2 if b = 14

Ans

wer

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45 Find (b/2)2 if b = -12A

nsw

er

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46 Complete the square to form a perfect square trinomial

x2 + 18x + ?

Ans

wer

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47 Complete the square to form a perfect square trinomial

x2 - 6x + ?

Ans

wer

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Solving quadratic equations by completing the square:

Step 1 - Write the equation in the form x 2 + bx = c

Step 2 - Find (b ÷ 2)2

Step 3 - Complete the square by adding (b ÷ 2)2 to both sides of the equation.

Step 4 - Factor the perfect square trinomial.

Step 5 - Take the square root of both sides

Step 6 - Write two equations, using both the positive and negative square root and solve each equation.

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Let's look at an example to solve:

x2 + 14x = 15 Step 1 - Already done!

(14 ÷ 2)2 = 49 Step 2 - Find (b÷2) 2

x2 + 14x + 49 = 15 + 49 Step 3 - Add 49 to both sides

(x + 7)2 = 64 Step 4 - Factor and simplify

x + 7 = ±8 Step 5 - Take the square root of both sides

x + 7 = 8 or x + 7 = -8 Step 6 - Write and solve two equationsx = 1 or x = -15

x2 + 14x = 15

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Another example to solve: x 2 - 2x - 2 = 0 x2 - 2x - 2 = 0 Step 1 - Write as x 2+bx=c +2 +2x2 - 2x = 2

(-2 ÷ 2)2 = (-1)2 = 1 Step 2 - Find (b÷2)2

x2 - 2x + 1 = 2 + 1 Step 3 - Add 1 to both sides

(x - 1)2 = 3 Step 4 - Factor and simplify

x - 1 = ± √3 Step 5 - Take the square root of both sides

x - 1 = √3 or x - 1 = -√3 Step 6 - Write and solve two equationsx = 1 + √3 or x = 1 - √3

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48 Solve the following by completing the square :

x2 + 6x = -5

A -5

B -2

C -1

D 5

E 2

Ans

wer

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x2 - 8x = 20

A -10

B -2

C -1

D 10

E 2

Ans

wer

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-36x=3x2 +108

A -6

B

C 0

D 6

E

Ans

wer

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Write as x 2+bx=c

Find (b÷2)2

Add 25/9 to both sides

Factor and simplify

Take the square root of both sides

Write and solve two equations

A more difficult example:

or

Slide 116 / 178


A

B

C

D

E

Ans

wer

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Solve Quadratic Equations by Using

the Quadratic Formula

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At this point you have learned how to solve quadratic equations by:· graphing· factoring· using square roots and · completing the square

Today we will be given a tool to solve ANY quadratic equation.

It ALWAYS works.

Many quadratic equations may be solved using these methods; however, some cannot be solved using any of these methods.

Slide 119 / 178

The Quadratic Formula

The solutions of ax 2 + bx + c = 0, where a ≠ 0, are:

"x equals the opposite of b, plus or minus the square root of b squared minus 4ac, all divided by 2a."

x=

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Example 1

2x 2 + 3x - 5 = 0

2x2 + 3x + ( -5 ) = 0

x = - b ± √ b 2 -4 ac 2 a

Write the Quadratic Formula

Substitute the values of a , b and c

Identify values of a , b and c

x = - 3 ± √ 3 2 -4( 2)( -5 ) 2( 2)

continued on next slide

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Simplify

Write as two equations

Solve each equationx = 1 or x = -5 2

x = -3 ± √9 - (-40) 4 x = -3 ± √49 4

= -3 ± 7 4

x = -3 + 7 4

x = -3 - 7 4

or

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Example 2 2x = x 2 - 3

Remember - In order to use the Quadratic Formula, the equation must be in standard form (ax 2 + bx +c = 0).

First, rewrite the equation in standard form.

-2x -2x

0 = x 2 + (-2x) + (-3)

x 2 + (-2x) + (-3) = 0 Flip the equation

Use only addition for standard form

Now you are ready to use the Quadratic Formula

2x = x 2 - 3

Solution on next slide

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x 2 + (-2x) + (-3) = 0

1x2 + ( -2 x) + ( -3 ) = 0

x = - b ± √ b 2 -4 ac 2 a




x = -( -2 ) ± √( -2 )2 -4( 1)( -3 ) 2( 1)

Continued on next slide

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Simplify


Solve each equationx = 3 or x = -1

x = 2 ± √4 - (-12) 2 x = 2 ± √16 2

= 2 ± 4 2

x = 2 + 4 2

x = 2 - 4 2

or

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52 Solve the following equation using the quadratic formula:

A -5

B -4

C -3

D -2

E -1

F 1

G 2

H 3

I 4

J 5

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A -5

B -4

C -3

D -2E -1

F 1

G 2

H 3I 4J 5

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A -5

B -4

C

D -2E -1

F 1

G 2

H

I 4

J 5

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Example 3

x 2 - 2x - 4 = 0

1x2 + ( -2 x) + ( -4 ) = 0

x = - b ± √ b 2 -4 ac 2 a




x = -( -2 ) ± √( -2 )2 -4( 1)( -4 ) 2( 1)


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Simplify


Use a calculator to estimate x

x = 2 ± √4 - (-16) 2 x = 2 ± √20 2 x = 2 + √20 2

x = 2 - √20 2

or

x ≈ 3.24 or x ≈ -1.24

x = 2 +2 √5 2

x = 2 - 2√5 2

or

x = 1 + √5 or x = 1 - √5

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55 Find the larger solution toA

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56 Find the smaller solution to

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The Discriminant


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x = - b ± √ b 2 -4 ac 2 a

Discriminant - the part of the equation under the radical sign in a quadratic equation.

b 2 - 4 ac is the discriminant

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ax2 + bx + c = 0

The discriminant, b2 - 4ac, or the part of the equation under the radical sign, may be used to determine the number of real solutions there are to a quadratic equation.

If b 2 - 4ac > 0 , the equation has two real solutionsIf b 2 - 4ac = 0 , the equation has one real solutionIf b 2 - 4ac < 0 , the equation has no real solutions

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Remember:

The square root of a positive number has two solutions.

The square root of zero is 0.

The square root of a negative number has no real solution.

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√4 = ± 2

Example

(2) (2) = 4 and (-2)(-2) = 4

So BOTH 2 and -2 are solutions

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What is the relationship between the discriminant of a quadratic and its graph?

Discriminant

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Discriminant

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Discriminant

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57 What is value of the discriminant of 2x2 - 3x + 5 = 0 ?

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58 Find the number of solutions using the discriminant for 2x2 - 3x + 5 = 0

A 0

B 1

C 2

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59 What is value of the discriminant of x2 - 8x + 4 = 0 ?

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60 Find the number of solutions using the discriminant for x 2 - 8x + 4 = 0

A 0

B 1

C 2 Ans

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Application Problems


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Quadratic Equations and Applications

A sampling of applied problems that lend themselves to being solved by quadratic equations:

Number Reasoning

Free Falling ObjectsGeometry: Dimensions

Height of a Projectile

Distances

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The product of two consecutive negative integers is 1,122. What are the numbers?

Remember that consecutive integers are one unit apart, so the numbers are n and n + 1.

Multiplying to get the product:n(n + 1) = 1122 n2 + n = 1122 n2 + n - 1122 = 0 (n + 34)(n - 33) = 0

n = -34 and n = 33.

The solution is either -34 and -33 or 33 and 34, since the direction ask for negative integers -34 and -33 are the correct pair.

→ STANDARD Form→ FACTOR

Number Reasoning

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PLEASE KEEP THIS IN MIND

When solving applied problems that lead to quadratic equations, you might get a solution that does not satisfy the physical constraints of the problem.

For example, if x represents a width and the two solutions of the quadratic equations are -9 and 1, the value -9 is rejected since a width must be a positive number.

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61 The product of two consecutive even integers is 48. Find the smaller of the two integers.

Hint: x(x+2) = 48

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The product of two consecutive integers is 272. What are the numbers?

TRY THIS:

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The product of two consecutive even integers is 528. What is the smaller number?

62

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The product of two consecutive odd integers is 1 less than four times their sum. Find the two integers.

Let n = 1st number n + 2 = 2nd number

n(n + 2) = 4[n + (n + 2)] - 1n 2 + 2n = 4[2n + 2] - 1n 2 + 2n = 8n + 8 - 1n 2 + 2n = 8n + 7n 2 - 6n - 7 = 0(n - 7)(n + 1) = 0

n = 7 and n = -1Which one do you use? Or do you use both?

More of a challenge...

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If n = 7 then n + 2 = 9

If n = -1 then n + 2 = -1 + 2 = 1

7 x 9 = 4[7 + (7 + 2)] - 163 = 4(16) - 163 = 64 -163 = 63

(-1) x 1 = 4[-1 + (-1 + 2)] - 1-1 = 4[-1 + 1] - 1-1 = 4(0) - 1-1 = -1

We get two sets of answers.

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63 The product of a number and a number 3 more than the original is 418. What is the smallest value the original number can be?

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64 Find three consecutive positive even integers such that the product of the second and third integers is twenty more than ten times the first integer. Enter the value of the smaller even integer.


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65 When 36 is subtracted from the square of a number, the result is five times the number. What is the positive solution?

A 9

B 6

C 3

D 4


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66 Tamara has two sisters. One of the sisters is 7 years older than Tamara.The other sister is 3 years younger than Tamara. The product of Tamara’s sisters’ ages is 24. How old is Tamara?


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Distance Problems

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Two cars left an intersection at the same time, one heading north and one heading west. Some time later, they were exactly 100 miles apart. The car that headed north had gone 20 miles farther than the car headed west. How far had each car traveled?

Step 1 - Read the problem carefully.

Step 2 - Illustrate or draw your information. x+20

100

x

Example

Step 3 - Assign a variableLet x = the distance traveled by the car heading westThen (x + 20) = the distance traveled by the car heading northStep 4 - Write an equationDoes your drawing remind you of the Pythagorean Theorem? a2 + b2 = c2


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Step 5 - Solve a2 + b2 = c2

x2 + (x+20)2 = 1002

x2 + x2 + 40x + 400 = 10,000

2x2 + 40x - 9600 = 0

Square the binomial

Standard form.

2(x2 +20x - 4800) = 0 Factor

Divide each side by 2.x2 + 20x - 4800 = 0

Think about your options for solving the rest of this equation. Completing the square? Quadratic Formula?

x+20100

x


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Did you try the quadratic formula?

2x = -20 ±√400 - 4(1)(-4800)

x = -20 ±√19,6002

x = 60 or x = -80

Since the distance cannot be negative, discard the negative solution. The distances are 60 miles and 60 + 20 = 80 miles.

Step 6 - Check your answers.

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67 Two cars left an intersection at the same time,one heading north and the other heading east. Some time later they were 200 miles apart. If the car heading east traveled 40 miles farther, how far did the car traveling north go?

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Geometry Applications

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The length of a rectangle is 6 inches more than its width. The area of the rectangle is 91 square inches. Find the dimensions of the rectangle.

Step 1 - Draw the picture of the rectangle. Let the width = x and the length = x + 6

Step 2 - Write the equation using theformula Area = length x width

Area Problem

x

x + 6

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Step 3 - Solve the equation

x( x + 6) = 91

x2 + 6x = 91

x2 + 6x - 91 = 0

(x - 7)(x + 13) = 0

x = 7 or x = -13

Since a length cannot be negative...

The width is 7 and the length is 13.

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68 The width of a rectangular swimming pool is 10 feet less than its length. The surface area of the pool is 600 square feet. What is the pool's width?

Hint: (L)(L - 10) = 600.

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69 A square's length is increased by 4 units and its width is increased by 6 units. The result of this transformation is a rectangle with an area that 195 square units. Find the area of the original square.

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70 The rectangular picture frame below is the same width all the way around. The photo it surrounds measures 17" by 11". The area of the frame and photo combined is 315 sq. in. What is the length of the outer frame?

length

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71 The area of the rectangular playground enclosure at South School is 500 square meters. The length of the playground is 5 meters longer than the width. Find the dimensions of the playground, in meters. [Only an algebraic solution will be accepted.]


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72 Jack is building a rectangular dog pen that he wishes to enclose. The width of the pen is 2 yards less than the length. If the area of the dog pen is 15 square yards, how many yards of fencing would he need to completely enclose the pen?

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Free Falling Objects Problems

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73 A person walking across a bridge accidentally drops an orange in the river below from a height of 40 ft. The function gives the orange's approximate height above the water, in feet, after t seconds. In how many seconds will the orange hit the water? (Round to the nearest tenth) Hint: when it hits the water it is at 0. A

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74 Greg is in a car at the top of a roller-coaster ride. The distance, d, of the car from the ground as the car descends is determined by the equation d = 144 - 16t2 where t is the number of seconds it takes the car to travel down to each point on the ride. How many seconds will it take Greg to reach the ground?

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75 The height of a golf ball hit into the air is modeled by the equation h = -16t2 + 48t, where h represents the height, in feet, and t represents the number of se conds that have passed since the ball was hit. What is the height of the ball after 2 seconds?A 16 ftB 32 ft

C 64 ft

D 80 ftFrom the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

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Height of Projectiles

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76 A skyrocket is shot into the air. It's altitude in feet, h, after t seconds is given by the function . What is the rocket's maximum altitude?

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77 A rocket is launched from the ground and follows a parabolic path represented by the equation y = -x 2 + 10x. At the same time, a flare is launched from a height of 10 feet and follows a straight path represented by the equation y = -x + 10. Using the accompanying set of axes, graph the equations that represent the paths of the rocket and the flare, and find the coordinates of the point or points where the paths intersect.


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slide 3 / 178macintodd.net/.../02/quadratics...slides-per-page.pdfslide 7 / 178 characteristics of...

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