Download x + 2 - mrhubbard

Table of Contents
Key Terms
Identify Quadratic Functions
Explain Characteristics of Quadratic Functions
Graphing Quadratic Functions
Transforming Quadratic Functions
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
The Discriminant
Solving Application Problems
Key Terms
Return to Table of
Contents
Axis of symmetry: The vertical line that divides a
parabola into two symmetrical halves
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
Max
(+ a)
(- a)
Min
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.
Quadratic Function: Any function that can be
written in the form y = ax2 + bx + c. Where a, b
and c are real numbers and a does not equal 0.
Vertex: The highest or lowest point on a
parabola.
Zero of a Function: An x value that makes the
function equal zero.
Identifying Quadratic
Functions
Return to Table of
Contents
Any quadratic function that can be written in the form
y = ax2 + bx + c (standard form)
Where a, b, and c are real numbers and
a≠0
Examples
Question: Is y = 7x + 9x2- 4 written in standard form?
Answer: No
Question: Is y = 0.5x2 + 7x written in standard form?
Answer: Yes
Characteristics
of Quadratic
Equations
Return to Table of
Contents
A quadratic equation is an equation of the form
ax2 + bx + c = 0 , where a is not equal to 0.
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?
Practice writing quadratic equations in standard form:
(Reduce if possible.)
Write 2x2 = x + 4 in standard form:
2x2 - x - 4 = 0
Write 3x = -x2 + 7 in standard form:
x2 + 3x - 7 = 0
Write 6x2 - 6x = 12 in standard form:
x2 - x - 2 = 0
Write 3x - 2 = 5x in standard form:
Not a quadratic
equation
Characteristics of Quadratic Functions
1. Standard form is y = ax2 + bx + c, where a ≠ 0.
2. The graph of a quadratic is a parabola, a u-shaped figure.
3. The parabola will open upward or downward.
downward
→
← upward
4. 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
5. The domain of a quadratic function is all real numbers.
6. 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
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
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
8. The x-intercepts are the points at which a parabola intersects the xaxis. These points are also known as zeroes, roots or solutions and
solution sets. Each quadratic function will have two, one or no real
x-intercepts.
1
True or False: The vertex is the highest or lowest value on the parabola.
True
False
2
If a parabola opens upward then...
A
a>0
B
a<0
C
a=0
3
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
4
What is the equation of the axis of symmetry of the parabola shown in the diagram below?
20
A
B
x=2
C
x = 4.5
D
5
x = -0.5
18
16
14
12
10
8
6
4
2
5 6 7
x = 13
8 9 10
1 2 3 4
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?
5
A
y=5
B
y = -5
C
x=5
D
x = -5
6
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
7
The equation y = x2 + 3x − 18 is graphed on the set of axes below.
Based on this graph, what are the roots
equation x2 + 3x − 18 = 0?
A
−3 and 6
B
0 and −18
C
3 and −6
D
3 and −18
of the
8
The equation y = − x2 − 2x + 8 is graphed 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
Graphing Quadratic Functions
Return to Table of
Contents
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
Step 1 - Find the Axis of Symmetry
What is the
Axis of Symmetry?
Axis of Symmetry
Step 1 - Find the Axis of Symmetry
Graph y = 3x2 – 6x + 1
Formula:
a=3
b = -6
x = - (- 6) = 6 = 1
2(3)
6
The axis of symmetry
is x = 1.
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
y = 3(1)2 + -6(1) + 1
y=3-6+1
y = -2
Vertex = (1 , -2)
a = 3, b = -6 and c = 1
Step 3 - Find y intercept.
y- intercept
What is the
y-intercept?
Graph y = 3x2 – 6x + 1
Step 3 - Find y- intercept.
The y- intercept is always the c value,
because x = 0.
y = ax2 + bx + c
c=1
y = 3x2 – 6x + 1
The y-intercept is 1 and
the graph passes through
(0,1).
Graph y = 3x2 – 6x + 1
Step 4 - Find two points
Choose different values of x and
plug in to find points.
Let's pick x = -1 and x = -2
y = 3x2 – 6x + 1
y = 3(-1)2 – 6(-1) + 1
y = 3 +6 + 1
y = 10
(-1,10)
Step 4 - Find two points
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)
Step 5 - Graph the axis of symmetry, the vertex, the point
containing the
y-intercept and two other points.
Step 6 - Reflect the points across the axis of symmetry.
Connect the points with a smooth curve.
(4,25)
9
What is the axis of symmetry for
y = x2 + 2x - 3 (Step 1)?
A
1
B
-1
What is the vertex for y = x2 + 2x - 3
(Step 2)?
10
A
(-1,-4)
B
(1,-4)
C
(-1,4)
11
What is the y-intercept for
y = x2 + 2x - 3 (Step 3)?
A
-3
B
3
12
What is an equation of the axis of symmetry of the parabola represented by
4?
A
x=3
B
y=3
C
x=6
D
y=6
y = −x2 + 6x −
Graph
Graph
Graph
On the set of axes below, solve the following system of equations graphically for all values of x and y .
y = -x2 - 4x + 12
y = -2x + 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
On the set of axes below, solve the following system of equations graphically for all values of x and y.
y = x2 − 6x + 1
y + 2x = 6
13
The graphs of the equations y = x2 + 4x - 1 and y + 3 = x are drawn on the same set of
axes. At which point do the graphs intersect?
A
(1,4)
B
(1,–2)
C
(–2,1)
D
(–2,–5)
Transforming Quadratic
Functions
Return to Table of
Contents
The quadratic parent function is f(x) = x2. The graph of all other quadratic
functions are transformations of the graph of f(x) = x2.
x
x2
-3
9
-2
4
-1
1
0
0
1
1
2
4
3
9
The quadratic parent function is f(x) = x2. How is
f(x) = x2 into f(x) = 2x2?
x
2
-3
18
-2
8
-1
2
0
0
1
2
2
8
3
18
The quadratic parent function is f(x) = x2. How is
f(x) = x2 into f(x) = .5x2?
x
0.5
-3
4.5
-2
2
-1
0.5
0
0
1
0.5
2
2
3
4.5
What does "a" do in
?
How does a>0 effect the parabola?
How does a<0 effect the parabola?
What does "a" do in
?
How does your conclusion about "a" change as a changes?
What does "a" do in
?
If a > 0, the graph opens up.
If a < 0, the graph opens down.
If the absolute value of a is > 1, then the graph of the function is narrower than
the graph of the parent function.
If the absolute value of a is < 1, then the graph of the function is wider than the
graph of the parent function.
Consider an "absolute value" number line to compare a parabola to parent function.
narrower
wider
0
1
parent function
2
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
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
16
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
17
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
18
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
What does "c" do in
?
What does "c" do in
?
"c" moves the graph up or down the same value as "c".
"c" is the y- intercept.
19
Without graphing, what is the y- intercept of the the given parabola?
20
Without graphing, what is the y- intercept of the the given parabola?
21
Without graphing, what is the y- intercept of the the given parabola?
22
Without graphing, what is the y- intercept of the the given parabola?
23
Choose all that apply to the following quadratic:
A
opens up
E
y-intercept of y = -4
B
opens down
F
y-intercept of y = -2
C
wider than parent function
G
y-intercept of y = 0
D
narrower than parent function
H
y-intercept of y = 2
I
y-intercept of y = 4
J
y-intercept of y = 6
24
Choose all that apply to the following quadratic:
A
opens up
E
y-intercept of y = -4
B
opens down
F
y-intercept of y = -2
C
wider than parent function
G
y-intercept of y = 0
D
narrower than parent function
H
y-intercept of y = 2
I
y-intercept of y = 4
J
y-intercept of y = 6
25
The diagram below shows the graph of y = −x2 − c.
Which diagram shows the graph of y = x2 − c?
A
B
C
D
Quadratic Equations
Finding Zeros (x- intercepts)
Solve Quadratic Equations by
Graphing
Return to Table of
Contents
Vocabulary
Every quadratic function has a related quadratic equation.
A quadratic equation is used to find the zeroes of a quadratic function. When a
function intersects the x-axis its y-value is zero.
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
One way to solve a quadratic equation in standard form is find the
zeros of the related function by graphing.
A zero is the point at which the parabola intersects the x-axis.
A quadratic may have one, two or no zeros.
How many zeros do the parabolas have?
zeros?
No zeroes
(doesn't cross the "x" axis)
What are the values of the
2 zeroes;
x = -1 and x=3
1 zero;
x=1
One way to solve a quadratic equation in standard form is to find the zeros or
x-intercepts of the related function.
Solve a quadratic equation by graphing:
Step 1 - Write the related function.
Step 2 - Graph the related function.
Step 3 - Find the zeros (or x intercepts) of the related function.
( ___, 0)
Step 1 - Write the Related Function
2x2 - 18 = 0
2x2 - 18 = y
y = 2x2 + 0x - 18
Step 2 - Graph the Function
y = 2x2 + 0x – 18
Use the same six step process for graphing
The axis of symmetry is x = 0.
The vertex is (0,-18).
Find the y intercept -- It is -18.
Find two other points (2,-10) and (3,0)
Two Points
Find two more points on the same side of
the Axis of Symmetry and as the point
containing the
y-intercept.
Step 2 - Graph the Function
y = 2x2 + 0x – 18
Graph the points and reflect them across the axis of symmetry.
x=0
●
●
(3,0)
●
●
(2,-10)
●
(0,-18)
Step 3 - Find the zeros
y = 2x2 + 0x – 18
Solve the equation by graphing the related function.
x=0
●
●
The zeros appear to be 3 and -3.
(3,0)
●
●
(2,-10)
●
(0,-18)
Step 3 - Find the zeros
y = 2x2 + 0x – 18
Substitute 3 and -3 for x in the quadratic equation.
Check 2x2 – 18 = 0
2(3)2 – 18 = 0
2(9) - 18 = 0
18 - 18 = 0
0=0
The zeros are 3 and -3.
2(-3)2 – 18 = 0
2(9) - 18 = 0
18 - 18 = 0
0=0
Solve the equation by graphing the related function.
26
-12x + 18 = -2x2
Step 1: Which of these is the related function?
A
y = -2x2 - 12x + 18
B
y = 2x2 - 12x + 18
C
y = -2x2 + 12x - 18
27
What is the axis of symmetry?
y = -2x2 + 12x - 18
A
-3
B
3
C
4
D
-5
Formula:
-b
2a
y = -2x2 + 12x - 18
28
What is the vertex?
A
(3,0)
B
(-3,0)
C
(4,0)
D
(-5,0)
y = -2x2 + 12x - 18
29
What is the y- intercept?
A
(0,0)
B
(0, 18)
C
(0, -18)
D
(0, 12)
30
If two other points are (5,-8) and (4,-2), what does the graph of y = -2x2
+ 12x - 18 look like?
A
B
C
D
y = -2x2 + 12x - 18
31
What is(are) the zero(s)?
A
-18
B
4
C
3
D
-8
Solve Quadratic Equations by
Factoring
Return to Table of
Contents
Solving Quadratic Equations
by 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 x2 + 9x + 18, look for factors whose sum is 9.
Sum
Factors of 18
1 and 18
19
2 and 9
11
3 and 6
9
x2 + 9x + 18 = (x + 3)(x + 6)
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.
Remember the FOIL method for multiplying binomials
1. Multiply the First terms
(x + 3)(x + 2)
x x = x2
2. Multiply the Outer 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) = x2 + 2x + 3x + 6 = x2 + 5x + 6
F
O
I
L
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
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
or
22 + 4(2) - 12 = 0
4 + 8 - 12 = 0
0=0
Use "FUSE" !
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.
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.
x2 - 12x + 36 = 0
(x - 6)(x - 6) = 0
Factor the trinomial
using the FOIL method.
x-6 =0
+6 +6
x=6
Use the Zero property
62 + 36 = 12(6)
36 + 36 = 72
Substitue found value into
original equation
72 = 72
Equal - problem solved!
Example 3: Solve x2 - 16x + 64 = 0
(x - 8)(x - 8) = 0
Factor the trinomial
using the FOIL method.
x-8=0
+8 +8
x=8
Use the Zero property
82 - 16(8) + 64 = 0
Substitue found value into
original equation
64 - 128 + 64 = 0
-64 + 64 = 0
0=0
Equal - problem solved!
32
Solve
A
-7
F
3
B
-5
G
5
C
-3
H
6
D
-2
I
7
E
2
J
15
33
Solve
A
-7
F
3
B
-5
G
5
C
-3
H
6
D
-2
I
7
E
2
J
15
34
Solve
A
-12
F
3
B
-4
G
4
C
-3
H
6
D
-2
I
8
E
2
J
12
35
Solve
A
-7
F
0
B
-5
G
5
C
-3
H
6
D
35
I
7
E
12
J
37
36
Solve
3/4
A
-3/4
F
1/2
B
-1/2
G
4/3
C
-4/3
H
D
-2
I
-3
E
2
J
3
What is the solution set of the equation x2 - 5x = 0?
37
A
{0,–5}
B
{0,5}
C
{0}
D
{5}
38
The solution set for the equation x2 - 5x = 6 is
A
{1,–6}
B
{2,–3}
C
{–1,6}
D
{–2,3}
39
What is the positive solution of the equation 4x2 - 36 = 0?
Which equation has roots of −3 and 5?
40
A
x2 + 2x − 15 = 0
B
x2 − 2x − 15 = 0
C
x2 + 2x + 15 = 0
D
x2 − 2x + 15 = 0
Solve Quadratic Equations Using
Square Roots
Return to Table of
Contents
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
Solve for z:
z = ± √49
z = ±7
z² = 49
The solution set is 7 and -7
A quadratic equation of the form x2 = c can be solved using the Square Root
Property.
Example: Solve 4x2 = 20
4x2 = 20
4
4
x2 = 5
x = ±√5
The solution set is √5 and -√5
Divide both sides by 4 to
isolate x²
Solve 5x² = 20 using the square root method:
5x2 = 20
5
5
x2 = 4
x =√4 or x = -√4
x = ±2
Solve (2x - 1)² = 20 using the square root method.
or
41
When you take the square root of a real number, your answer will
always be positive.
True
False
42
If x2 = 16, then x =
A
4
B
2
C
-2
D
26
E
-4
43
If y2 = 4, then y =
A
4
B
2
C
-2
D
26
E
-4
44
If 8j2 = 96, then j =
A
B
C
D
E
45
If 4h2 -10= 30, then h =
A
B
C
D
E
46
If (3g - 9)2 + 7= 43, then g =
A
B
C
D
E
Solving Quadratic Equations by
Completing the Square
Return to Table of
Contents
Form a perfect square trinomial with lead coefficient of 1
x2 + bx +c where c = (b/2)2
Find the value that completes the square.
16
100
64
1
47
Find (b/2)2 if b = 14
48
Find (b/2)2 if b = -12
49
Complete the square to form a perfect square trinomial
x2 + 18x + ?
50
Complete the square to form a perfect square trinomial
x2 - 6x + ?
Solving quadratic equations by completing the square:
Step 1 - Write the equation in the form x2 + 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.
x2 + 14x = 15
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
(x + 7)2 = 64
x + 7 = ±8
sides
Step 4 - Factor and simplify
Step 5 - Take the square
x + 7 = 8 or x + 7 = -8
equations
x = 1 or x = -15
root of both sides
Step 6 - Write and solve two
Another example to solve: x2 - 2x - 2 = 0
x2 - 2x - 2 = 0
+2 +2
x2 - 2x = 2
Step 1 - Write as x2+bx=c
(-2 ÷ 2)2 = (-1)2 = 1
x2 - 2x + 1 = 2 + 1
Step 2 - Find (b÷2)2
Step 3 - Add 1 to both sides
(x - 1)2 = 3
Step 4 - Factor and simplify
x - 1 = ± √3
Step 5 - Take the square
x - 1 = √3 or x - 1 = -√3
equations
x = 1 + √3 or x = 1 - √3
root of both sides
Step 6 - Write and solve two
51
Solve the following by completing the square :
x2 + 6x = -5
A
-5
B
-2
C
-1
D
5
E
2
52
Solve the following by completing the square :
x2 - 8x = 20
A
-10
B
-2
C
-1
D
10
E
2
53
Solve the following by completing the square :
-36x=3x2 +108
A
-6
B
C
0
D
6
E
A more difficult example:
Write as x2+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
or
54
Solve the following by completing the square :
A
B
C
D
E
Solve Quadratic Equations by Using
the Quadratic Formula
Return to Table of
Contents
At this point you have learned how to solve quadratic equations by:
graphing
factoring
using square roots and
completing the square
Many quadratic equations may be solved using these methods; however,
some cannot be solved using any of these methods.
Today we will be given a tool to solve ANY quadratic equation.
It ALWAYS works.
The Quadratic Formula
The solutions of ax2 + bx + c = 0, where a ≠ 0, are:
x = -b ± √b2 - 4ac
2a
"x equals the opposite of b, plus or minus the square root of b squared
minus 4ac, all divided by 2a."
Example 1
2x2 + 3x - 5 = 0
2x2 + 3x + (-5) = 0
Identify values of a, b and c
x = -b ± √b2 -4ac
2a
x = -3 ± √32 -4(2)(-5)
2(2)
Write the Quadratic Formula
Substitute the values of a, b and c
continued on next slide
x = -3 ± √9 - (-40)
Simplify
4
x = -3 ± √49
= -3 ± 7
4
4
x = -3 + 7
4
x = 1 or x = -5
2
or
x = -3 - 7
4
Write as two equations
Solve each equation
Example 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
Use only addition for standard form
0 = x2 + (-2x) + (-3)
x2 + (-2x) + (-3) = 0
Flip the equation
Now you are ready to use the Quadratic Formula
Solution on next slide
x2 + (-2x) + (-3) = 0
1x2 + (-2x) + (-3) = 0
Identify values of a, b and c
x = -b ± √b2 -4ac
2a
Write the Quadratic Formula
x = -(-2) ± √(-2)2 -4(1)(-3)
2(1)
Substitute the values of a, b and c
Continued on next slide
x = 2 ± √4 - (-12)
Simplify
2
x = 2 ± √16
=2±4
2
2
x=2+4
2
x = 3 or x = -1
x=2-4
or
2
Write as two equations
Solve each equation
55
Solve the following equation using the quadratic formula:
A
-5
F
1
B
-4
G
2
C
-3
H
3
D
-2
I
4
E
-1
J
5
56
Solve the following equation using the quadratic formula:
A
-5
F
1
B
-4
G
2
C
-3
H
3
D
-2
I
4
E
-1
J
5
57
Solve the following equation using the quadratic formula:
A
-5
F
1
B
-4
G
2
H
C
D
-2
E
-1
I
4
J
5
Example 3
x2 - 2x - 4 = 0
1x2 + (-2x) + (-4) = 0
Identify values of a, b and c
x = -b ± √b2 -4ac
2a
x = -(-2) ± √(-2)2 -4(1)(-4)
2(1)
Write the Quadratic Formula
Substitute the values of a, b and c
Continued on next slide
Simplify
x = 2 ± √4 - (-16)
2
x = 2 ± √20
2
x = 2 + √20
2
x = 2 - √20
2
or
Write as two equations
x = 2 +2 √5
2
x = 1 + √5
or
x = 2 - 2√5
2
or x = 1 - √5
x ≈ 3.24 or x ≈ -1.24
Use a calculator to estimate x
58
Find the larger solution to
59
Find the smaller solution to
The Discriminant
Return to Table of
Contents
Discriminant - the part of the equation under the radical sign
in a quadratic equation.
x = -b ± √b2 -4ac
2a
b2 - 4ac is the discriminant
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 b2 - 4ac > 0, the equation has two real solutions
If b2 - 4ac = 0, the equation has one real solution
If b2 - 4ac < 0, the equation has no real solutions
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.
Example
√4 = ± 2
(2) (2) = 4 and (-2)(-2) = 4
So BOTH 2 and -2 are solutions
What is the relationship between the discriminant of a quadratic and its graph?
Discriminant
What is the relationship between the discriminant of a quadratic and its graph?
Discriminant
What is the relationship between the discriminant of a quadratic and its graph?
Discriminant
60
What is value of the discriminant of 2x2 - 3x + 5 = 0 ?
Find the number of solutions using the discriminant for 2x2 - 3x + 5 = 0
61
A
0
B
1
C
2
62
What is value of the discriminant of x2 - 8x + 4 = 0 ?
Find the number of solutions using the discriminant for x2 - 8x + 4 = 0
63
A
0
B
1
C
2
Application Problems
Return to Table of
Contents
Quadratic Equations and Applications
A sampling of applied problems that lend themselves to being solved by
quadratic equations:
Number Reasoning
Distances
Geometry: Dimensions
Free Falling Objects
Height of a Projectile
Number Reasoning
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.
→ STANDARD Form
The solution is either -34→
and
-33 or 33 and 34, since the direction ask for negative
FACTOR
integers -34 and -33 are the correct pair.
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.
64
The product of two consecutive even integers is 48. Find the smaller of the two
integers.
Hint: x(x+2) = 48
TRY THIS:
The product of two consecutive integers is 272. What are the numbers?
65
Answer?
The product of two consecutive even integers is 528. What are the numbers?
(Enter the smaller factor in Senteo)
66
The product of two consecutive positive even integers is 168. Find the larger
of the numbers.
67
Answer?
The product of two consecutive odd integers is 255. What are the numbers?
(Enter the larger factor in Senteo)
More of a challenge...
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)] - 1
n2 + 2n = 4[2n + 2] - 1
n2 + 2n = 8n + 8 - 1
n2 + 2n = 8n + 7
n2 - 6n - 7 = 0
(n - 7)(n + 1) = 0
n = 7 and n = -1
Which one do you use? Or do you use both?
If n = 7 then n + 2 = 9
7 x 9 = 4[7 + (7 + 2)] - 1
63 = 4(16) - 1
63 = 64 -1
63 = 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.
68
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?
69
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 into your senteo.
70
Three brothers have ages that are consecutive even integers. The prod- uct of the first and third
boys’ ages is 20 more than twice the second boy’s age. Find the age of each of the three boys.
Enter the age of the youngest boy in your senteo.
71
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
72
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?
Distance Problems
Example
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 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.
100
x+20
Step 3 - Assign a variable
Let x = the distance traveled by the car heading west
Then (x + 20) = the distance traveled by the car heading north
x
Step 4 - Write an equation
Does your drawing remind you of the Pythagorean Theorem? a2 +
b2 = c2
Continued on next slide
Step 5 - Solve a2 + b2 = c2
100
x+20
x2 + (x+20)2 = 1002
x
x2 + x2 + 40x + 400 = 10,000
Square the binomial
2x2 + 40x - 9600 = 0
Standard form.
2(x2 +20x - 4800) = 0
Factor
x2 + 20x - 4800 = 0
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
Did you try the quadratic formula?
x = -20 ±√400 - 4(1)(-4800)
2
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.
73
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
travel travel 40 miles farther, How far did the car traveling north go?
Geometry Applications
Area Problem
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 the
formula Area = length x width
x+6
x
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.
74
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.
75
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.
76
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
77
The rectangular fish pond below is the same width all the
way around. You have enough concrete to cover 72 sq. ft.
for a walkway. What should the width of the walkway be?
78
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.]
79
An architect is designing a museum entranceway in the shape of a parabolic arch
represented by the equation y = -x2 + 20x, where 0 x 20 and all dimensions are
expressed in feet. On the accompanying set of axes, sketch a graph of the arch and
determine its maximum height, in feet.
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.
80
A rocket is launched from the ground and follows a parabolic path represented by the equation y = -x2 + lOx.
At the same time, a flare is launched from a height of 10 feet and follows a straight path rep resented 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 from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
81
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?
Free Falling Objects Problems
82
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
seconds.
In how many seconds will the orange hit the water? (Round to the
nearest tenth)
83
An acorn drops from a tree branch 70 ft. above the ground. The function
gives the height of the acorn after
seconds. Graph the function. At about what
time does the acorn hit the ground?
84
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?
85
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?
A
16 ft
B
32 ft
C
64 ft
D
80 ft
Height of Projectiles
86
A skyrocket is shot into the air. It's altitude
by the function
What is the rocket's maximum altitude?
in feet after
seconds is given
87
You are trying to dunk a basketball. You need to jump 2.5 ft. in the air in
order to dunk the ball. The height that your feet are above the ground is
given by the function
What is the maximum height your feet will be above the ground?