# 7 2 Solving Quadratic Equations

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```					7.2 Solving Quadratic
Equations
Algebra 2
Mrs. Spitz
Spring 2007
Objectives

   Solve quadratic equations by graphing,
 Find the equation of the axis of symmetry
and
 find the coordinates of the vertex of the
   solve quadratic equations by factoring
Assignment

   pp. 319-320 #6-45 odd

   A quadratic function is a function that
can be described by an equation of the
form y = ax2 + bx + c, where a ≠ 0.
Generalities
   The path of an object
when it is thrown or
dropped is called the
TRAJECTORY of the
object. A bouncing
object like the tennis
ball in the photo will
have a trajectory in a
general shape called
a parabola.
Generalities
   Equations such as y =
6x – 0.5x2 and y = x2 –
4x +1 describe a type of
function known as a
functions have common
characteristics. For
instance, they all have
the general shape of a
parabola.
Generalities
The table and graph can be used to illustrate other common
characteristics of quadratic functions. Notice the matching values
in the y-column of the table.
6
y = x2 – 4x + 1
x        x2 – 4x + 1        y
-1      (-1)2 – 4(1) + 1     6             4

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

2       (2)2 – 4(2) + 1     -3
3       (3)2 – 4(3) + 1     -2                                      5

4       (4)2 – 4(4) + 1      1
-2

5       (5)2   – 4(5) + 1    6
(2, -3)
-4
Generalities
Notice in the y-column of the table, -3 does not have a matching value. Also
notice that -3 is the y-coordinate of the lowest point of the graph. The
point (2, -3) is the lowest point, or minimum point, of the graph of y = x2 –
4x + 1.
6
y = x2 – 4x + 1
x          x2 – 4x + 1           y
-1        (-1)2 – 4(1) + 1        6                4

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

2        (2)2 – 4(2) + 1         -3
3        (3)2 – 4(3) + 1         -2                                       5

4        (4)2 – 4(4) + 1         1
-2

5        (5)2   – 4(5) + 1       6
(2, -3)
-4
Maximum/minimum points

 For the graph of y = 6x – 0.5x2, the point
(6, 18) is the highest point, or maximum
point. The maximum point or minimum
point of a parabola is also called the
vertex of the parabola.
 The graph of a quadratic function will
have a minimum point or a maximum,
BUT NOT BOTH!!!
Axis of Symmetry

 The vertical line containing the vertex of
the parabola is also called the axis of
symmetry for the graph. Thus, the
equation of the axis of symmetry for the
graph of y = x2 – 4x + 1 is x = 2
 In general, the equation of the axis of
symmetry for the graph of a quadratic
function can be found by using the rule
following.
Equation of the Axis of Symmetry

   The equation of
the axis of
symmetry for the        b
graph of
x
y = ax2 + bx + c,
where a ≠ 0, is
2a
Ex. 1: Find the equation of the axis of symmetry and the
coordinates of the vertex of the graph of y = x2 – x – 6. Then
use the information to draw the graph.

   First, find the axis of       NOTE: for
symmetry.
y = x2 – x – 6
b
x                       a = 1 b = -1 c = -6
2a
1
x  (      )
2 1
1
x
2
Ex. 1: Find the equation of the axis of symmetry and the
coordinates of the vertex of the graph of y = x2 – x – 6. Then
use the information to draw the graph.

1 2 1
   Next, find the vertex.            y  ( )  6
Since the equation of                  2   2
the axis of symmetry is x
1 1
= ½ , the x-coordinate of
the vertex must be ½ .
  6
You can find the y-
4 2
coordinate by                       1 2 24
substituting ½ for x in y           
= x2 – x – 6 .                      4 4 4
25
The point ( ½, -25/4) is              
the vertex of the graph.                   4
This point is a minimum.
Generalities
The table and graph can be used to illustrate other common
characteristics of quadratic functions. Notice the matching values
in the y-column of the table.                 2

y = x2 – x – 6
x         x2 – x – 6           y
-2       (-2)2 – (-2) – 6       0                                  5

-1       (-1)2 – (-1) – 6       -4            -2
x=½
0         (0)2 – (0) – 6        -6
1         (1)2 – (1) – 6        -6            -4

2         (2)2 – (2) – 6        -4
3         (3)2 – (3) - 6        0             -6

This point is a minimum!
-8       ½, -25/4)

The solution of a quadratic equation in one variable x can be solved
or checked graphically with the following steps:

STEP 1     Write the equation in the form ax 2 + bx + c = 0.

STEP 2     Write the related function y = ax 2 + bx + c.

STEP 3     Sketch the graph of the function y = ax 2 + bx + c.

The solutions, or roots, of ax 2 + bx + c = 0 are the x-intercepts.
Checking a Solution Using a Graph

1 2
Solve     x = 8 algebraically. Check your solution graphically.
2

SOLUTION
1 2           Write original equation.
x = 8
2

x 2 = 16     Multiply each side by 2.

x= 4      Find the square root of each side.

CHECK    Check these solutions using a graph.
Checking a Solution Using a Graph

CHECK Check these solutions using a graph.

1   Write the equation in the form ax 2 + bx + c = 0

1 2
x =8        Rewrite original equation.
2
1 2
x –8=0         Subtract 8 from both sides.
2

2   Write the related function y = ax2 + bx + c.

y = 1 x2 – 8
2
Checking a Solution Using a Graph

CHECK Check these solutions using a graph.

2   Write the related function
– 4, 0    4, 0
y = ax2 + bx + c.

y = 1 x2 – 8
2

1
3   Sketch graph of y = x2 – 8.
2
The x-intercepts are  4, which
agrees with the algebraic solution.
Solving an Equation Graphically

Solve x 2 – x = 2 graphically.

SOLUTION
1       Write the equation in the form ax 2 + bx + c = 0
x2 – x = 2       Write original equation.
x2 – x – 2 = 0    Subtract 2 from each side.
(x-2)(x+1)=0        Factor and set equal to zero.
x–2=0                 x+1=0
x=2                                Solve. These are your x-intercepts.
x = -1
2    Write the related function y = ax2 + bx + c.
y = x2 – x – 2
Solving an Equation Graphically

2   Write the related function y = ax2 + bx + c.

y = x2 – x – 2

– 1, 0   2, 0

3   Sketch the graph of the function
y = x2 – x – 2

From the graph, the x-intercepts
appear to be x = –1 and x = 2.
Solving an Equation Graphically

From the graph, the x-intercepts             – 1, 0   2, 0
appear to be x = –1 and x = 2.

CHECK

You can check this by substitution.

Check x = –1:               Check x = 2:
x2 – x = 2                 x2 – x = 2
?                         ?
(–1) 2 –   (–1) = 2               22 – 2 = 2
1+1=2                      4–2=2

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