5.2 Solving Quadratic
Equations
Algebra 2
Learning Targets

I can solve quadratic equations by
graphing,
Find the equation of the axis of symmetry
and
 find the coordinates of the vertex of the
graph of a quadratic function


I can solve quadratic equations by
factoring
Definition of a Quadratic Function

A quadratic function is a function that
can be described by an equation of the
form y = ax2 + bx + c, where a ≠ 0.
Generalities


Equations such as y =
6x – 0.5x2 and y = x2 –
4x +1 describe a type of
function known as a
quadratic function.
Graphs of quadratic
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
x
x2 – 4x + 1
y
-1
(-1)2 – 4(1) + 1
6
0
(0)2 – 4(0) + 1
1
1
(1)2 – 4(1) + 1
-2
2
(2)2 – 4(2) + 1
-3
3
(3)2 – 4(3) + 1
-2
4
(4)2 – 4(4) + 1
1
5
(5)2
– 4(5) + 1
y = x2 – 4x + 1
4
x=2
2
5
-2
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
x
x2 – 4x + 1
y
-1
(-1)2 – 4(1) + 1
6
0
(0)2 – 4(0) + 1
1
1
(1)2 – 4(1) + 1
-2
2
(2)2 – 4(2) + 1
-3
3
(3)2 – 4(3) + 1
-2
4
(4)2 – 4(4) + 1
1
5
(5)2
– 4(5) + 1
y = x2 – 4x + 1
4
x=2
2
5
-2
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
graph of
y = ax2 + bx + c,
where a ≠ 0, is
b
x
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
symmetry.
b
x
2a
1
x  (
)
2 1
1
x
2

NOTE: for
y = x2 – x – 6
a = 1 b = -1 c = -6
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.

Next, find the vertex.
Since the equation of
the axis of symmetry is x
= ½ , the x-coordinate of
the vertex must be ½ .
You can find the ycoordinate by
substituting ½ for x in y
= x2 – x – 6 .
The point ( ½, -25/4) is
the vertex of the graph.
This point is a minimum.
1 2 1
y  ( )  6
2
2
1 1
  6
4 2
1 2 24
  
4 4 4
25

4
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
x
x2 – x – 6
y
-2
(-2)2 – (-2) – 6
0
-1
(-1)2 – (-1) – 6
-4
0
(0)2 – (0) – 6
-6
1
(1)2 – (1) – 6
-6
2
(2)2 – (2) – 6
-4
3
(3)2 – (3) - 6
0
y = x2 – x – 6
5
x=½
-2
-4
-6
This point is a minimum!
-8
½, -25/4)
Solving Quadratic Equations Graphically
SOLVING QUADRATIC EQUATIONS USING GRAPHS
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
Solve
1 2
x = 8 algebraically. Check your solution graphically.
2
SOLUTION
1 2
x = 8
2
CHECK
Write original equation.
x 2 = 16
Multiply each side by 2.
x= 4
Find the square root of each side.
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
2
1 2
x –8=0
2
2
Rewrite original equation.
Subtract 8 from both sides.
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
y = ax2 + bx + c.
–4, 0
y = 1 x2 – 8
2
3
1
Sketch graph of y = x2 – 8.
2
The x-intercepts are  4, which
agrees with the algebraic solution.
4, 0
Solving an Equation Graphically
Solve x 2 – x = 2 graphically.
Check your solution algebraically.
SOLUTION
Write the equation in the form ax 2 + bx + c = 0
1
x2 – x = 2
x2 – x – 2 = 0
(x-2)(x+1)=0
x+1=0
x = -1
x–2=0
x=2
2
Write original equation.
Subtract 2 from each side.
Factor and set equal to zero.
Solve. These are your x-intercepts.
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
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.
2, 0
Solving an Equation Graphically
From the graph, the x-intercepts
appear to be x = –1 and x = 2.
– 1, 0
CHECK
You can check this by substitution.
Check x = –1:
Check x = 2:
x2 – x = 2
x2 – x = 2
?
?
(–1) = 2
22 – 2 = 2
1+1=2
4–2=2
(–1) 2 –
2, 0
Pair-share

P 260 #24-46 even