algebra i - center for teaching &...
TRANSCRIPT
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• Characteristics of Quadratic Equations
• Graphing Quadratic Equations
• Transforming Quadratic Equations
• Solve Quadratic Equations by Graphing
• Solve Quadratic Equations by Factoring
• Solve Quadratic Equations Using Square Roots
• Solve Quadratic Equations by Completing the Square
• Solve Quadratic Equations by Using the Quadratic Formula
• Key Terms
• Solving Application Problems
• The Discriminant
Table of ContentsClick on the topic to go to that section
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Axis of symmetry: The vertical line thatdivides a parabola intotwo symmetrical halves
Axis of Symmetry
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Maximum: The y-value of thevertex if a < 0 and the parabola opens downward
Minimum: The y-value of thevertex if a > 0 and theparabola opens upward
Parabola: The curve resultof graphing aquadratic equation
(+ a) Max
Min
(- a)
Parabolas
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Quadratic Equation: An equation that can be written in the standard form ax 2 + bx + c = 0. Where a, b and c are real numbers and a does not = 0.
Vertex: The highest or lowest point ona parabola.
Zero of a Function: An x value that makes the function equal zero.
Quadratics
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A quadratic equation is an equation of the form ax2 + bx + c = 0 , where a is not equal to 0.
Quadratics
The form ax2 + 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: (Simplify if possible.)
Write 2x2 = x + 4 in standard form:
Writing Quadratic Equations
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2 Write 6x2 - 6x = 12 in standard form:
A. 6x2 - 6x -12 = 0
B. x2 - x - 2 = 0
C. -x2 + x + 2 = 0
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3 Write 3x - 2 = 5x in standard form:
A. 2x + 2 = 0
B. -2x - 2 = 0
C. not a quadratic equation
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← upward→downward
The graph of a quadratic is a parabola, a u-shaped figure.
The parabola will open upward or downward.
Characteristics of Quadratic Functions
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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
Characteristics of Quadratic Functions
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The domain of a quadratic function is all real numbers.
Characteristics of Quadratic Functions
D = Reals
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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,∞ )
Characteristics of Quadratic Functions
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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 (–∞ ,10]
Characteristics of Quadratic Functions
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7. 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 of the form
x=2
Characteristics of Quadratic Functions
x = –b2a
x = –(–8) 2(2)
= 2 y = 2x2 – 8x + 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
Characteristics of Quadratic Functions
x=2x = –b2a
x = –(–8) 2(2)
= 2 y = 2x2 – 8x + 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 knownas zeroes, roots or solutions and solution sets. Each quadratic equation will have two, one or no real x-intercepts.
Characteristics of Quadratic Functions
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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
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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 = 1B 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.
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6 The equation y = x 2 + 3x − 18 isgraphed on the set of axes below.
−3 and 60 and −183 and −6 3 and −18
A B C D
Based on this graph, what are the roots of the equation x2 + 3x − 18 = 0?
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.
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C
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7 The equation y = − x 2 − 2x + 8 isgraphed on the set of axes below.
Based on this graph, what are the roots of the equation −x2 − 2x + 8 = 0?
A 8 and 0
B 2 and –4
C 9 and –1
D 4 and –2
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.
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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 = 3B y = 3C x = 6D y = 6
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.
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9 The height, y, of a ball tossed into the air can be represented 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
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.
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5 6 7 8 9 10
18 16 14 12 10
8642
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.
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 = 4.5D x = 13 A
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B
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x x2
-3 9-2 4-1 10 01 12 43 9
The quadratic parent equation is y = x2. The graph of all other quadratic equations are transformations of the graph of y= x2.
Quadratic Parent Equation
y = x2
y = – – x223
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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
Quadratic Parent Equation
y = 2x2
y = x2
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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?
Quadratic Parent Equation
y = x2
y = – x212
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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 ?What Does "A" Do?
y = x2 y = –x2
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What does "a" also do in y =ax2 + bx +c ?
How does your conclusion about "a" changeas "a" changes?
What Does "A" Do?
y = x2y = – x212
y = 3x2
y = –1x2 y = –3x2 y = – – x212
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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 ?
What Does "A" Do?
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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
wery = .3x2
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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
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y = –4x2
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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
y = –2x2 + 100x + 45
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14 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
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y = – – x223
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15 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
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B
y = – – x275
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What does "c" do in y = ax2 + bx + c ?
What Does "C" Do?
y = x2 + 6
y = x2 + 3
y = x2
y = x2 – 2
y = x2 – 5
y = x2 – 9
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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 ?
What Does "C" Do?
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18 Without graphing, what is the y- intercept of the the given parabola?
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y = –3x2 + 13x – 9
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20 Choose all that apply to the following quadratic:
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opens up
opens down
wider than parent function
narrower than parent
function
A
B
C
D
y-intercept of y = –4
y-intercept of y = –2
y-intercept of y = 0
y-intercept of y = 2
y-intercept of y = 4
y-intercept of y = 6
A
B
C
D
E
F
f(x) = –.7x2 –4
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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
E y-intercept of y = –4
F y-intercept of y = –2
G y-intercept of y = 0
H y-intercept of y = 2I y-intercept of y = 4
J y-intercept of y = 6
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f(x) = – – x2 –6x43
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22 The diagram below shows the graph of y = −x 2 − c.
A B C D
Which diagram shows the graph of y = x2 − c?
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.
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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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Axis of Symmetry
Axis of SymmetryStep 1 - Find Axis of Symmetry
What is the Axis of Symmetry?
Teac
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the center of a parabola. This line divides the
graph into two perfect halves.
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Step 1 - Find Axis of Symmetry
Graph y = 3x2 – 6x + 1
Formula:a = 3b = –6x = – (–6) = 6 = 1 2(3) 6
x = –––b2a
The axis of symmetryis x = 1.
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y = 3(1)2 + –6(1) + 1y = 3 – 6 + 1y = –2Vertex = (1,–2)
Step 2 - Find Vertex
Step 2 - Find the vertex by substituting the value of x(the axis of symmetry) into the equation to get y.
y = 3x2 – 6x + 1
a = 3, b = –6 and c = 1
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Step 3 - Find y interceptWhat is the y-intercept?
y- intercept
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The point where the line passes through they-axis. This occurs
when the x-value is 0.
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The y- intercept is always the c value, because x = 0.
c = 1
The y-intercept is 1 and the graph passes through (0,1).
y = ax2 + bx + c
y = 3x2 – 6x + 1
Step 3 - Find y intercept
Graph y = 3x2 – 6x + 1
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Choose different values of x and plug in to find points.
Step 4 - Find Two More Points
Find two more points on the parabola.
Graph y = 3x2 – 6x + 1
Let's pick x = –1 and x = –2
y = 3x2 – 6x + 1
y = 3(–1)2 – 6(–1) + 1
y = 3 + 6 + 1y = 10
(–1,10)
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Step 4 - Find Two More Points (continued)Graph y = 3x2 – 6x + 1
y = 3x2 – 6x + 1
y = 3(–2)2 – 6(–2) + 1
y = 3(4) + 12 + 1
y = 25
(–2, 25)
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Step 5 - Graph the Axis of Symmetry
Graph the axis of symmetry, the vertex, the point containing the y-intercept and two other points.
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(4,25)
Step 6 - Reflect the Points
Reflect the points across the axis of symmetry. Connect the points with a smooth curve.
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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)
Graphy= x2 + 2x – 3
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Find the Zeros
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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No zeroes(doesn't crossthe "x" axis)
2 zeroes; x = -1 and x=3
1 zero;x=1
Find the ZerosHow many zeros do the parabolas have?
What are the values of the zeros?
click click click
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Review
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
To solve a quadratic equation by graphing follow the6 step process we already learned.
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Which of these is in standard form?
y = 2x2 – 12x + 18
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Solve the equation by graphing.
–12x + 18 = –2x2
y = –2x2 – 12x + 18
y = –2x2 + 12x – 18
B
A
C
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What is the y- intercept?
A (0, 0)
B (0, 18)
C (0, –18)
D (0, 12)
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C
y = –2x2 + 12x – 18
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A B
C D
If two other points are (5, –8) and (4 ,–2),what doesthe graph of y = –2x2 + 12x – 18 look like?
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31 y = –2x2 + 12x – 18
What is(are) the zero(s)?
A –18
B 4
C 3
D –8
click for graph of answer
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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
Solving Quadratic Equationsby Factoring
x2 + 9x + 18 = (x + 3)(x + 6)
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When c is positive, it's 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.
Solving Quadratic Equationsby Factoring
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4. Multiply the Last terms (x + 3)(x + 2) 3 2 = 6
3. Multiply the Inner terms (x + 3)(x + 2) 3 x = 3x
2. Multiply the Outer terms (x + 3)(x + 2) x 2 = 2x
1. Multiply the First terms (x + 3)(x + 2) x x = x2
F O I L
Remember the FOIL method for multiplying binomials
Solving Quadratic Equationsby Factoring
(x + 3)(x + 2) = x2 + 2x + 3x + 6 = x2 + 5x + 6
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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
Zero Product Property
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Example 1: Solve x2 + 4x – 12 = 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" !
Zero Product Property
Factor the trinomial using the FOIL method.
Use the Zero property
Substitue found value into original equation
Equal - problem solved! The solutions are -6 and 2.
(x + 6) (x – 2) = 0
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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.
Zero Product Property
Factor the trinomial using the FOIL method.
Use the Zero property
Substitue found value into original equation
Equal - problem solved!
x2 – 12x + 36 = 0
(x – 6)(x – 6) = 0 x – 6 = 0 +6 +6 x = 6
62 + 36 = 12(6)
36 + 36 = 72
72 = 72
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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
Zero Product Property
Factor the trinomial using the FOIL method. Use the Zero property
Substitue found value into original equation
Equal - problem solved!
42 – 16(4) + 48 = 0 16 – 64 + 48 = 0 –48+48 = 0 0 = 0
122 – 16(12) + 48 = 0 144 –192 + 48 = 0 –48 + 48 = 0 0 = 0
–48
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37 Which equation has roots of −3 and 5?
A x2 + 2x − 15 = 0B x2 − 2x − 15 = 0C x2 + 2x + 15 = 0 D x2 − 2x + 15 = 0 A
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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 = c or x = – c
written as:
x = ± c
Square Root Method
√ √
√
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The solution set is and –
A quadratic equation of the form x2 = c can be solved using the Square Root Property.
Example: Solve 4x2 = 20
x = ±
Square Root Method
4x2 = 20 4 4 x2 = 5
Divide both sides by 4 to isolate x²
5√ 5√
5√
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5x2 = 20 5 5
x2 = 4
x = or x = – x = ± 2
4 4
Square Root Method
Solve 5x² = 20 using the square root method:
√ √
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2x – 1 = 202x – 1 = (4)(5)2x – 1 = 2 52x = 1 + 2 5 1 + 2 5x = 2
Solve (2x – 1)² = 20 using the square root method. or
Square Root Method
2x – 1 = – 202x – 1 = – (4)(5)2x – 1 = –2 52x = 1 – 2 5 1 – 2 5x = 2
solution: x = 1 ± 2 5
2
√
√√√√
√
√√√√
√click
click click
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38 When you take the square root of a real number, your answer will always be positive.
True
False
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False
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x2 + 8x + ___
x2 + 20x + 100
x2 – 16x + 64
x2 – 2x + 1
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.
(b/2)2
8/2 = 4
42 = 16
ax2+bx+c
Find the Missing Value of "C"
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Solving Quadratic Equationsby 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
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 equations
x = 1 or x = –15
Solving Quadratic Equationsby Completing the Square
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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
Solving Quadratic Equationsby Completing the Square
√
√ √
√ √
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50 Solve the following by completing the square :
–36x = 3x 2 + 108
A –6
B 6C 0D 6E – 6
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A
116
x – = ±5 43 3
10x 3
x2 – = –1
Write as x 2+bx=c
Find (b÷2)2
Add 25/9 to both sidesFactor and simplify
A more difficult example:Solve3x2 – 10x = –3
3x2 10x = –3 3 3 3
–
10 3
÷ 2 = x = = –10 1 –5 25 3 2 3 9)( ( ) ( )
2 2 2
10x 25 25 3 9 9
x2 – + = –1 +
x – =5 163 9)(
2
√ Take the square root of both sides
Answ
er
x – =
x = 3
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– 254
51 Solve the following by completing the square:
Ans
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4x2 – 7x – 2 = 0
– 14
A
B
C
D
E
14
– 254
–
2
119
x = –b ± √b2 – 4ac 2a
Discriminant - the part of the equation under the radical sign in a quadratic equation.
The Discriminant
b2 – 4ac 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.
The Discriminant
If b2 – 4ac > 0, the equation has two real solutionsIf b2 – 4ac = 0, the equation has one real solutionIf b2 – 4ac < 0, the equation has no real solutions
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The Discriminant
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.
123
What is the relationship between the discriminant of a quadratic and its graph?
Discriminant(8)2 – 4(1)(10) = 64 – 40 = 24 (–6)2 –4(3)(–4) = 36 + 48 = 84
The Discriminant
y = x2 – 8x + 10 y = 3x2 + 8x – 4
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What is the relationship between the discriminant of a quadratic and its graph?
The Discriminant
Discriminant(–4)2 – 4(2)(2) = 16 – 16 = 0 (6)2 –4(1)(9) = 36 – 36 = 0
y = 2x2 – 4x + 2 y = x2 + 6x + 9
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What is the relationship between the discriminant of a quadratic and its graph?
Discriminant
(5)2 – 4(1)(9) = 25 – 36 = –11 (–3)2 –4(3)(4) = 9 – 48 = –39
The Discriminant
y = x2 + 5x + 9 y = 3x2 – 3x + 4
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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.
Solve Any Quadratic Equation
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"x equals the opposite of b, plus or minus the square root of b squared minus 4ac, all divided by 2a."
The Quadratic Formula
The solutions of ax2 + bx + c = 0, where a ≠ 0, are:
x = –b ± b2 – 4ac√2a
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x = –3 ± √32 –4(2)(–5) 2(2)
x = –b ± √b2 –4ac 2a
continued on next slide
Write the Quadratic Formula
Identify values of a, b and c
Substitute the values of a, b and c
2x2 + 3x + (–5) = 0
2x2 + 3x – 5 = 0
Example 1
The Quadratic Formula
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x = –3 – 7 4
= –3 ± 7 4
x = –3 ± √49 4
x = –3 ± √9 – (–40) 4
x = –3 + 7 4 or
The Quadratic Formula
Simplify
Write as two equations
Solve each equationx = 1 or x = –5 2
135
Solution on next slide
The Quadratic FormulaExample 2 2x = x2 – 3
Remember - In order to use the Quadratic Formula, the equation must be in standard form (ax2 + bx +c = 0).
First, rewrite the equation in standard form.
2x = x2 – 3–2x –2x
0 = x2 + (-2x) + (–3)
x2 + (–2x) + (–3) = 0
Use only addition for standard form
Flip the equation
Now you are ready to use the Quadratic Formula
136
x = –(–2) ± √(–2)2 –4(1)(–3) 2(1)
x = –b ± √b2 –4ac 2a
Continued on next slide
The Quadratic Formulax2 + (–2x) + (–3) = 0
1x2 + (–2x) + (–3) = 0 Identify values of a, b and c
Write the Quadratic Formula
Substitute the values of a, b and c
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x = 2 ± √162
= 2 ± 42
x = 2 ± √4 – (–12)2a
Simplify
x = 2 ± 42
or x = 2 - 42
x = 3 or x = –1
Write as two equations
Solve each equation
The Quadratic Formula
138
56 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
Ans
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x2 – 5x + 4 = 0
139
57 Solve the following equation using the quadratic formula:
A –5
B –4
C –3
D –2
E –1
F 1
G 2
H 3I 4J 5
Ans
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x2 = x + 20
140
–3 2
58 Solve the following equation using the quadratic formula:
A –5
B –4
C
D –2
E –1
F 1
G 2
H
I 4
J 5
Ans
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2x2 + 12 = 11x
32
141
x = -b ± √b2 -4ac
2a
Continued on next slide
The Quadratic FormulaExample 3
x2 – 2x – 4 = 0
1x2 + (–2x) + (–4) = 0 Identify values of a, b and c
Write the Quadratic Formula
Substitute the values of a, b and c
x = –(–2) ± √(–2)2 –4(1)(–4) 2(1)
142
The Quadratic Formula
x = 2 ± √202
x = 2 ± √4 – (–16)2
Simplify
x = 2 ± 2√52
Write as two equationsor x = 2 - 2√52
orx = 2 ± √202
x = 2 - √202
x = 1 + √5 or x = 1 – √5
x ≈ 3.24 or x ≈ –1.24 Use a calculator to estimate x
146
A sampling of applied problems that lend themselves to being solved by quadratic equations:
Number Reasoning
Free Falling Objects
DistancesGeometry: Dimensions
Height of a Projectile
Quadratic Equations and Applications
147
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 directionask for negative integers –34 and –33 are the correct pair.
→ STANDARD Form→ FACTOR
Number Reasoning
148
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.
Application Problems
149
61 The product of two consecutive even integers is 48. Find the smaller of the two integers.
Hint: x(x+2) = 48
Ans
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150
The product of two consecutive integers is 272.
What are the numbers?
TRY THIS:
Application Problems
152
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
More of a challenge...
n(n + 2) = 4[n + (n + 2)] – 1n2 + 2n = 4[2n + 2] – 1n2 + 2n = 8n + 8 – 1n2 + 2n = 8n + 7n2 – 6n - 7 = 0(n – 7)(n + 1) = 0
n = 7 and n = –1
Which one do you use?Or do you use both?
153
More of a challenge...
If n = 7 then n + 2 = 9
7 x 9 = 4[7 + (7 + 2)] – 163 = 4(16) – 163 = 64 – 163 = 63
If n = –1 then n + 2 = –1 + 2 = 1
(–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.
154
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?
Ans
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155
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 th e value of the smaller even integer.
Ans
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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.
156
65 When 36 is subtracted from the square of a number, the result is five times the number. What is the positive solution?
A 9B 6C 3D 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
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157
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?
Ans
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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.
158
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 Continued on next slide
Distance Problems
159
Step 5 - Solve a2 + b2 = c2
x2 + (x+20)2 = 1002
x2 + x2 + 40x + 400 = 10,000
2x2 + 40x – 9600 = 0
2(x2 +20x – 4800) = 0
x2 + 20x – 4800 = 0
100x+20
x
Square the binomial
Standard form
Factor
Divide each side by 2
Think about your options for solving the rest of this equation. Completing the square? Quadratic Formula?
Continued on next slide
Distance Problems
160
x = –20 ±√400 – 4(1)(–4800) 2
Distance ProblemsDid you try the quadratic formula?
x = –20 ±√19,600 2
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.
161
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?
Ans
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162
x
x + 6
Geometry Applications
Area ProblemThe 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
163
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.
Geometry Applications
164
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. Ans
wer
Click to reveal hint
165
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.
Ans
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166
length
x
x
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?
Answ
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167
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.]
Answ
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From the New York State Education Department. Office of Assessment Policy,
Development and Administration. Internet. Available fromwww.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
168
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?
Answ
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170
73
Answ
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A person walking across a bridge accidentally drops an orange in the river below from a height of 40 ft. The function h = –16t2 + 40 gives the orange's approximate height h above the water, in feet, after t seconds. In how many t seconds will the orange hit the water? (Round to the nearest tenth.)
Hint: when it hits the water it is at 0.
171
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 – 16t 2 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?
Answ
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172
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 seconds that have passed since the ball was hit. What is the height of the ball after 2 seconds?
Answ
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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.
A 16 ft
B 32 ft
C 64 ft
D 80 ft
174
76 A skyrocket is shot into the air. It's altitude in feet, h, after t seconds is given by the functionh = –16 t2 + 128 t.
What is the rocket's maximum altitude?
Answ
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175
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.
From the New York State Education Department. Office of Assessment Policy,
Development and Administration. Internet. Available fromwww.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
Answ
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