Graph quadratic equations.

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Objectives Students will learn how to

• Graph quadratic equations.
• Complete the square to graph quadratic equations.
• Use the Vertex Formula to graph quadratic
  equations.
• Solve a Quadratic Equation by factoring and
  square root property.
• Complete the square to solve quadratics.
• Use the quadratic formula to solve quadratics.
• Solve for a Specified Variable
• Understand the properties of the Discriminant

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A function ? is a quadratic function if
where a, b, and c are real numbers, with a ? 0.
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Simplest Quadratic
range 0, ?)
y
x ?(x)
2 4
1 1
0 0
1 1
2 4
x
domain (-?, ?)
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Simplest Quadratic

• Parabolas are symmetric with respect to a line.
  The line of symmetry is called the axis of
  symmetry of the parabola. The point where the
  axis intersects the parabola is the vertex of the
  parabola.

Opens up
Vertex
Axis
Axis
Vertex
Opens down
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The graph of g(x) ax2 is a parabola with vertex
at the origin that opens up if a is positive and
down if a is negative. The width of the graph of
g(x) is determined by the magnitude of a. The
graph of g(x) is narrower than that of ?(x) x2
if ?a?gt 1 and is broader (wider) than that of
?(x) x2 if ?a?lt 1. By completing the square,
any quadratic function can be written in vertex
form the graph of F(x) is the same as the
graph of g(x) ax2 translated ?h?units
horizontally (to the right if h is positive and
to the left if h is negative) and translated ?k?
units vertically (up if k is positive and down if
k is negative).
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Example 1

• GRAPHING QUADRATIC FUNCTIONS

Graph the function. Give the domain and range.
a.
x ?(x)
1 3
0 2
1 5
2 6
3 5
4 2
5 3
Solution
Domain (-?, ?)
Range 6, ?)
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Example 1

• GRAPHING QUADRATIC FUNCTIONS

Graph the function. Give the domain and range.
b.
Solution
Domain (-?, ?)
Range (?, 0
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Example 1

• GRAPHING QUADRATIC FUNCTIONS

Graph the function. Give the domain and range.
c.
Solution
Domain (-?, ?)
Range (?, 3
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Example 2

• GRAPHING A PARABOLA BY COMPLETING THE SQUARE

Graph by completing
the square and locating the vertex.
Solution Express x2 6x 7 in the form (x h)2
k by completing the square.

Complete the square.
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Example 2

• GRAPHING A PARABOLA BY COMPLETING THE SQUARE

Graph by completing
the square and locating the vertex.
Solution Express x2 6x 7 in the form (x
h)2 k by completing the square.

Add and subtract 9.
Regroup terms.
Factor simplify.
This form shows that the vertex is (3, 2)
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Example 2

• GRAPHING A PARABOLA BY COMPLETING THE SQUARE

Graph by completing
the square and locating the vertex.
Solution
Find additional ordered pairs that satisfy the
equation. Use symmetry about the axis of the
parabola to find other ordered pairs. Connect to
obtain the graph.
Domain is (-?, ?)
Range is 2, ?)
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Example 3

• GRAPHING A PARABOLA BY COMPLETING THE SQUARE

Graph by
completing the square and locating the vertex.
Solution To complete the square, the coefficient
of x2 must be 1.
Factor 3 from the first two terms.
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Example 3

• GRAPHING A PARABOLA BY COMPLETING THE SQUARE

Graph by
completing the square and locating the vertex.
Solution
Distributive property
Be careful here.
Factor simplify.
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Example 3

• GRAPHING A PARABOLA BY COMPLETING THE SQUARE

Graph by
completing the square and locating the vertex.
Solution
Factor simplify.
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Example 3

• GRAPHING A PARABOLA BY COMPLETING THE SQUARE

Graph by
completing the square and locating the vertex.
Solution Intercepts are good additional points
to find. Here is the y-intercept.

Let x 0.
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Example 3

• GRAPHING A PARABOLA BY COMPLETING THE SQUARE

Graph by
completing the square and locating the vertex.

• Solution The x-intercepts are found by setting
  ?(x) equal to 0 in the original equation.
  

Let ?(x) 0.
Multiply by 1 rewrite.
Factor.
Zero-factor property
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Example 3

• GRAPHING A PARABOLA BY COMPLETING THE SQUARE

The x intercepts or zeros are the solutions to
the quadratic equations
The y-intercept is 1
This x-intercept is 1/3.
This x-intercept is 1.
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Vertex Formula
The quadratic function defined by
?(x) ax2 bx c can be written as where
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• The graph of a quadratic function has the
  following characteristics. Vertex form
• It is a parabola with vertex (h, k) and the
  vertical line x h as axis of symmetry.
• It opens up if a gt 0 and down is a lt 0.
• It is broader than the graph of y x2 if ?a?lt 1
  and narrower if ?a?gt 1.
• The y-intercept is ?(0) c.

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Example 4

• FINDING THE AXIS,VERTEX, SOLUTIONS AND GRAPH OF A
  PARABOLA.

Find the axis, vertex, solutions and graph of the
parabola having equation ?(x) 2x2 4x 5 using
the vertex formula.
Solution Here a 2, b 4, and c 5. The axis
of the parabola is the vertical line

Axis is x -1
The vertex is ( 1, ?( 1)). Since ?( 1) 2(
1)2 4 ( 1) 5 3, the vertex is
( 1, 3).
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Example 4
FINDING THE AXIS,VERTEX, SOLUTIONS AND GRAPH OF A
PARABOLA.
No real solutions because the graph does not have
a value at y 0 the graph does not cross the
x-axis
X Y
-3 11
-2 5
-1 3
0 5
1 11
Axis x -1
This function has solutions at x3 and x6.
These values are also called zeros because the y
value is zero for x3 and x6
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If a and b are numbers with ab 0, then a 0 or
b 0 or both.
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Example 1

• USING THE ZERO-FACTOR PROPERTY

Solve
Solution
Standard form
Factor.
Zero-factor property.
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Example 1

• USING THE ZERO-FACTOR PROPERTY

Solve
Solution
Zero-factor property.
Solve each equation.
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Square-Root Property

• A quadratic equation of the form x2 k can also
  be solved by factoring

Subtract k.
Factor.
Zero-factor property.
Solve each equation.
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If x2 k, then
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Square-Root Property

• That is, the solution of

Both solutions are real if k gt 0, and both are
imaginary if k lt 0
If k 0, then this is sometimes called a double
solution.
If k lt 0, we write the solution set as
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Example 2

• USING THE SQUARE ROOT PROPERTY

Solve each quadratic equation.
a.
Solution
By the square root property, the solution set is
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Example 2

• USING THE SQUARE ROOT PROPERTY

Solve each quadratic equation.
b.
Solution
Since
the solution set of x2 - 25
is
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Example 2

• USING THE SQUARE ROOT PROPERTY

Solve each quadratic equation.
c.
Solution
Use a generalization of the square root property.
Generalized square root property.
Add 4.
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To solve ax2 bx c 0, by completing the
square Step 1 If a ? 1, divide both sides of
the equation by a. Step 2 Rewrite the equation
so that the constant term is alone on one side of
the equality symbol. Step 3 Square half the
coefficient of x, and add this square to both
sides of the equation. Step 4 Factor the
resulting trinomial as a perfect square and
combine like terms on the other side. Step 5 Use
the square root property to complete the solution.
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Example 3

• USING THE METHOD OF COMPLETING THE SQUARE a 1

Solve x2 4x 14 0 by completing the square.
Solution
Step 1 This step is not necessary since a 1.
Add 14 to both sides.
Step 2
Step 3
add 4 to both sides.
Step 4
Factor combine terms.
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Example 3

• USING THE METHOD OF COMPLETING THE SQUARE a 1

Solve x2 4x 14 0 by completing the square.
Solution
Step 4
Factor combine terms.
Step 5
Square root property.
Take both roots.
Add 2.
Simplify the radical.
The solution set is
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Example 4

• USING THE METHOD OF COMPLETING THE SQUARE a ? 1

Solve 9x2 12x 9 0 by completing the square.
Solution
Divide by 9. (Step 1)
Add 1. (Step 2)
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Example 4

• USING THE METHOD OF COMPLETING THE SQUARE a 1

Solve 9x2 12x 9 0 by completing the square.
Solution
Factor, combine terms. (Step 4)
Square root property
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Example 4

• USING THE METHOD OF COMPLETING THE SQUARE a 1

Solve 9x2 12x 9 0 by completing the square.
Solution
Square root property
Quotient rule for radicals
Add ?.
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Example 4

• USING THE METHOD OF COMPLETING THE SQUARE a 1

Solve 9x2 12x 9 0 by completing the square.
Solution
Add ?.
The solution set is
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The Quadratic Formula

• The method of completing the square can be used
  to solve any quadratic equation. If we start
  with the general quadratic equation, ax2 bx c
  0, a ? 0, and complete the square to solve this
  equation for x in terms of the constants a, b,
  and c, the result is a general formula for
  solving any quadratic equation. We assume that a
  gt 0.

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The solutions of the quadratic equation ax2 bx
c 0, where a ? 0, are
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Caution Notice that the fraction bar in
the quadratic formula extends under the b term
in the numerator.
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Example 5

• USING THE QUADRATIC FORMULA (REAL SOLUTIONS)

Solve x2 4x 2
Solution
Write in standard form.
Here a 1, b 4, c 2
Quadratic formula.
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Example 5

• USING THE QUADRATIC FORMULA (REAL SOLUTIONS)

Solve x2 4x 2
Solution
Quadratic formula.
The fraction bar extends under b.
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Example 5

• USING THE QUADRATIC FORMULA (REAL SOLUTIONS)

Solve x2 4x 2
Solution
The fraction bar extends under b.
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Example 5

• USING THE QUADRATIC FORMULA (REAL SOLUTIONS)

Solve x2 4x 2
Solution
Factor out 2 in the numerator.
Factor first, then divide.
Lowest terms.
The solution set is
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Example 6

• USING THE QUADRATIC FORMULA (NONREAL COMPLEX
  SOLUTIONS)

Solve 2x2 x 4.
Solution
Write in standard form.
Quadratic formula a 2, b 1, c 4
Use parentheses and substitute carefully to avoid
errors.
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Example 6

• USING THE QUADRATIC FORMULA (NONREAL COMPLEX
  SOLUTIONS)

Solve 2x2 x 4.
Solution
The solution set is
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Example 8

• SOLVING FOR A QUADRATIC VARIABLE IN A FORMULA

Solve for the specified variable. Use ? when
taking square roots.
a.
Solution
Goal Isolate d, the specified variable.
Multiply by 4.
Divide by ?.
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Example 8

• SOLVING FOR A QUADRATIC VARIABLE IN A FORMULA

Solve the specified variable. Use ? when taking
square roots.
a.
Solution
Divide by ?.
Square root property
See the Note following this example.
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Example 8

• SOLVING FOR A QUADRATIC VARIABLE IN A FORMULA

Solve the specified variable. Use ? when taking
square roots.
a.
Solution
Square root property
Rationalize the denominator.
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Example 8

• SOLVING FOR A QUADRATIC VARIABLE IN A FORMULA

Solve the specified variable. Use ? when taking
square roots.
a.
Solution
Rationalize the denominator.
Multiply numerators multiply denominators.
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Example 8

• SOLVING FOR A QUADRATIC VARIABLE IN A FORMULA

Solve the specified variable. Use ? when taking
square roots.
a.
Solution
Multiply numerators multiply denominators.
Simplify.
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Solving for a Specified Variable
Note In Example 8, we took both positive and
negative square roots. However, if the variable
represents a distance or length in an
application, we would consider only the positive
square root.
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Example 8

• SOLVING FOR A QUADRATIC VARIABLE IN A FORMULA

Solve the specified variable. Use ? when taking
square roots.
b.
Solution
Write in standard form.
Now use the quadratic formula to find t.
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Example 8

• SOLVING FOR A QUADRATIC VARIABLE IN A FORMULA

Solve the specified variable. Use ? when taking
square roots.
b.
Solution
a r, b s, and c k
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Example 8

• SOLVING FOR A QUADRATIC VARIABLE IN A FORMULA

Solve the specified variable. Use ? when taking
square roots.
b.
Solution
a r, b s, and c k
Simplify.
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The Discriminant

• The Discriminant The quantity under the radical
  in the quadratic formula, b2 4ac, is called the
  discriminant.

Discriminant
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The Discriminant
Discriminant Number of Solutions Type of Solution
Positive, perfect square Two Rational
Positive, but not a perfect square Two Irrational
Zero One (a double solution) Rational
Negative Two Nonreal complex
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Caution The restriction on a, b, and c is
important. For example, for the equation
the discriminant is b2 4ac 5 4 9, which
would indicate two rational solutions if the
coefficients were integers. By the quadratic
formula, however, the two solutions are
irrational numbers,
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Example 9

• USING THE DISCRIMINANT

Determine the number of distinct solutions, and
tell whether they are rational, irrational, or
nonreal complex numbers.
a.
Solution
For 5x2 2x 4 0, a 5, b 2, and c 4.
The discriminant is b2 4ac 22 4(5)( 4)
84 The discriminant is positive and not a perfect
square, so there are two distinct irrational
solutions.
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Example 9

• USING THE DISCRIMINANT

Determine the number of distinct solutions, and
tell whether they are rational, irrational, or
nonreal complex numbers.
b.
Solution
First write the equation in standard form as x2
10x 25 0. Thus, a 1, b 10, and c
25, and b2 4ac ( 10 )2 4(1)(25) 0 There
is one distinct rational solution, a double
solution.
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Example 9

• USING THE DISCRIMINANT

Determine the number of distinct solutions, and
tell whether they are rational, irrational, or
nonreal complex numbers.
c.
Solution
For 2x2 x 1 0, a 2, b 1, and c 1,
so b2 4ac (1)2 4(2)(1) 7. There are
two distinct nonreal complex solutions. (They
are complex conjugates.)