Quadratic Equations

1
Quadratic Equations
Chapter 16
2
Chapter Sections
16.1 Solving Quadratic Equations by the Square
Root Property 16.2 Solving Quadratic Equations
by Completing the Square 16.3 Solving Quadratic
Equations by the Quadratic Formula 16.4
Graphing Quadratic Equations in Two
Variables 16.5 Interval Notation, Finding
Domains and Ranges from Graphs and Graphing
Piecewise-Defined Functions
3
16.1

• Solving Quadratic Equations by the Square Root
  Property

4
Square Root Property

• We previously have used factoring to solve
  quadratic equations.
• This chapter will introduce additional methods
  for solving quadratic equations.
• Square Root Property
• If b is a real number and a2 b, then

5
Square Root Property
Example

• Solve x2 49

Solve 2x2 4
x2 2
Solve (y 3)2 4
y 3 ? 2 y 1 or 5
6
Square Root Property
Example

• Solve x2 4 0
• x2 ?4
• There is no real solution because the square root
  of ?4 is not a real number.

7
Square Root Property
Example

• Solve (x 2)2 25

x ?2 5 x ?2 5 or x ?2 5 x 3 or x
?7
8
Square Root Property
Example

• Solve (3x 17)2 28

9
16.2

• Solving Quadratic Equations by Completing the
  Square

10
Completing the Square
In all four of the previous examples, the
constant in the square on the right side, is half
the coefficient of the x term on the left. Also,
the constant on the left is the square of the
constant on the right. So, to find the constant
term of a perfect square trinomial, we need to
take the square of half the coefficient of the x
term in the trinomial (as long as the coefficient
of the x2 term is 1, as in our previous examples).
11
Completing the Square
Example

• What constant term should be added to the
  following expressions to create a perfect square
  trinomial?

• x2 10x
• add 52 25
• x2 16x
• add 82 64
• x2 7x

12
Completing the Square
Example

• We now look at a method for solving quadratics
  that involves a technique called completing the
  square.
• It involves creating a trinomial that is a
  perfect square, setting the factored trinomial
  equal to a constant, then using the square root
  property from the previous section.

13
Completing the Square

• Solving a Quadratic Equation by Completing a
  Square
• If the coefficient of x2 is NOT 1, divide both
  sides of the equation by the coefficient.
• Isolate all variable terms on one side of the
  equation.
• Complete the square (half the coefficient of the
  x term squared, added to both sides of the
  equation).
• Factor the resulting trinomial.
• Use the square root property.

14
Solving Equations
Example

• Solve by completing the square.
• y2 6y ?8
• y2 6y 9 ?8 9
• (y 3)2 1

y ?3 1 y
?4 or ?2
15
Solving Equations
Example

• Solve by completing the square.
• y2 y 7 0
• y2 y 7
• y2 y ¼ 7 ¼

16
Solving Equations
Example

• Solve by completing the square.
• 2x2 14x 1 0
• 2x2 14x 1
• x2 7x ½

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16.3

• Solving Quadratic Equations by the Quadratic
  Formula

18
The Quadratic Formula

• Another technique for solving quadratic equations
  is to use the quadratic formula.
• The formula is derived from completing the square
  of a general quadratic equation.

19
The Quadratic Formula

• A quadratic equation written in standard form,
  ax2 bx c 0, has the solutions.

20
The Quadratic Formula
Example

• Solve 11n2 9n 1 by the quadratic formula.
• 11n2 9n 1 0, so
• a 11, b -9, c -1

21
The Quadratic Formula
Example
x2 8x 20 0 (multiply both sides by
8) a 1, b 8, c ?20
22
The Quadratic Formula
Example

• Solve x(x 6) ?30 by the quadratic formula.
• x2 6x 30 0
• a 1, b 6, c 30

So there is no real solution.
23
The Discriminant

• The expression under the radical sign in the
  formula (b2 4ac) is called the discriminant.
• The discriminant will take on a value that is
  positive, 0, or negative.
• The value of the discriminant indicates two
  distinct real solutions, one real solution, or no
  real solutions, respectively.

24
The Discriminant
Example

• Use the discriminant to determine the number and
  type of solutions for the following equation.
• 5 4x 12x2 0
• a 12, b 4, and c 5
• b2 4ac (4)2 4(12)(5)
• 16 240
• 224
• There are no real solutions.

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Solving Quadratic Equations

• Steps in Solving Quadratic Equations
• If the equation is in the form (axb)2 c, use
  the square root property to solve.
• If not solved in step 1, write the equation in
  standard form.
• Try to solve by factoring.
• If you havent solved it yet, use the quadratic
  formula.

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Solving Equations
Example

• Solve 12x 4x2 4.
• 0 4x2 12x 4
• 0 4(x2 3x 1)
• Let a 1, b -3, c 1

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Solving Equations
Example

• Solve the following quadratic equation.

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16.4

• Graphing Quadratic Equations in Two Variables

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Graphs of Quadratic Equations
We spent a lot of time graphing linear equations
in chapter 3. The graph of a quadratic equation
is a parabola. The highest point or lowest point
on the parabola is the vertex. Axis of symmetry
is the line that runs through the vertex and
through the middle of the parabola.
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Graphs of Quadratic Equations
Example
Graph y 2x2 4.
(2, 4)
(2, 4)
2
4
1
2
(1, 2)
(1, 2)
0
4
1
2
(0, 4)
2
4
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Intercepts of the Parabola
Although we can simply plot points, it is helpful
to know some information about the parabola we
will be graphing prior to finding individual
points. To find x-intercepts of the parabola, let
y 0 and solve for x. To find y-intercepts of
the parabola, let x 0 and solve for y.
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Characteristics of the Parabola
If the quadratic equation is written in standard
form, y ax2 bx c, 1) the parabola opens
up when a gt 0 and opens down when a lt 0.
To find the corresponding y-coordinate, you
substitute the x-coordinate into the equation and
evaluate for y.
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Graphs of Quadratic Equations
Example
Graph y 2x2 4x 5.
(1, 7)
(2, 5)
(0, 5)
3
1
(1, 1)
(3, 1)
5
2
1
7
0
5
1
1
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16.5

• Interval Notation, Finding Domain and Ranges from
  Graphs, and Graphing Piecewise-Defined Functions

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Domain and Range

• Recall that a set of ordered pairs is also called
  a relation.
• The domain is the set of x-coordinates of the
  ordered pairs.
• The range is the set of y-coordinates of the
  ordered pairs.

36
Domain and Range
Example

• Find the domain and range of the relation (4,9),
  (4,9), (2,3), (10, 5)
• Domain is the set of all x-values, 4, 4, 2, 10
• Range is the set of all y-values, 9, 3, 5

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Domain and Range
Example
Find the domain and range of the function graphed
to the right. Use interval notation.
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Domain and Range
Example
Find the domain and range of the function graphed
to the right. Use interval notation.
39
Domain and Range
Example
Find the domain and range of the following
relation.

• Input (Animal)
• Polar Bear
• Cow
• Chimpanzee
• Giraffe
• Gorilla
• Kangaroo
• Red Fox

• Output (Life Span)
• 20
• 15
• 10
• 7

40
Domain and Range
Example continued
Domain is Polar Bear, Cow, Chimpanzee, Giraffe,
Gorilla, Kangaroo, Red Fox Range is 20, 15, 10,
7
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Graphing Piecewise-Defined Functions
Example
Graph each piece separately.
x f (x) 3x 1
0 1(closed circle)
1 4
2 7
x f (x) x 3
1 4
2 5
3 6
Values ? 0.
Values gt 0.
Continued.
42
Graphing Piecewise-Defined Functions
Example continued
x f (x) 3x 1
0 1(closed circle)
1 4
2 7
x f (x) x 3
1 4
2 5
3 6