Transcript Part 2- Solving Quadratic Equations by Graphing

9-2 Solving Quadratic Equations by
Graphing
Algebra 1
Glencoe McGraw-Hill
Linda Stamper
and JoAnn Evans
Factoring can be used to determine whether the graph of a
quadratic functions intersects the x-axis in one or two
points.
The graph intersects the x-axis when f(x) equals 0.
fx   x2  x  12 function
Factor.
Identify the
roots.
x2  x  12  0
related equation
x  3x  4   0
–12
x  3  0 or x  4  0 –3
4
x  4
x 3
1
The graph of the function intersects the x-axis two times.
y
fx   x2  x  12
y-intercept
x
y
-3
-6
-4

b
2a

0
1
49

2
4
2
-6
3
0
–4, 3
matchy
,
matchy
!
•
•
x
Use factoring to determine how many times the graph of
each function intersects the x-axis. Identify each root.
Remember: The graph intersects the x-axis when f(x) equals 0.
Example 1
fx   x2  3x  10
Two roots; -5, 2
Example 2
fx   x2  10 x  25
One root; 5
Example 3
2
fx   3x  5x  2
Example 4
fx   2x2  21x  11
2
Two roots; -1,  3
Two roots; -11,
1
2
In previous graphing to find solutions, the roots of the
equations were integers. Usually the roots of a quadratic
equation are not integers. In these cases, use estimation
to approximate the roots of the equation.
Solve x2 + 6x + 7 by graphing. If integral roots cannot be
found, estimate the roots by stating the consecutive
integers between which the roots lie.
Integral roots are roots that are integers (positive and
negative whole numbers).
y
x2  6x  7  0
x2  6x  7  y
x
y
-4
-1
-5

b
2a
2
3 2
-2
-1
-1
2
matchy
,
matchy
!
• •
•••
–5<x<-4, -2<x<-1
Notice that the value of the function changes from positive to
negative between x values of -5 and -4 and between -2 and -1.
The x-intercepts are between -5 and -4 and between -2 and -1. So
one root is between -5 and -4 and the other between -2 and -1.
x
Solve by graphing. If integral roots cannot be found,
estimate the roots by stating the consecutive integers
between which the roots lie.
Example 5 2x2  6x  3  0
Example 6
 x2  3x  4  0
Example 7
x2  2x  5  0
Example 8
x2  4x  5  0
Example 9
2x2  5  10x
Example 5
2x2  6x  3  0
2x2  6x  3  y
x
y
0
-3
-1
-7
matchy
,
matchy
!
-1.5 -7.5
-2
-7
-3
-3
–4 < x < -3, 0 < x < 1
Example 6
 x2  3x  4  0
 x2  3x  4  y

b
2a
x
y
0
-4
1
-2
2
-2
3
-4
3
7

4
2
y
matchy
,
matchy
!
no real solution
•••
• •
x
Example 7
x2  2x  5  0
x2  2x  5  y

b
2a
x
y
-3
-2
-2
-5
-1
-6
0
-5
1
-2
–5 < x < -4,
y
matchy
,
matchy
!
1<x<2
• •
•••
x
y
Example 8
x2  4x  5  0
x2  4x  5  y
x

b
2a
y
0
-5
1
-8
2 -9
3 -8
4 -5
matchy
,
matchy
!
-1, 5
• •
•••
x
y
Example 9
2x2  5  10x
2x2  10x  5  0
2x2  10x  5  y

b
2a
x
y
1
-3
2
-7
5
2

3
-7
4
-3
15
2
x
matchy
,
matchy
!
0 < x < 1,
• •
•••
4<x<5
Solve by graphing. If integral roots cannot be found,
estimate the roots by stating the consecutive integers
between which the roots lie.
Example 5 2x2  6x  3  0
–4 < x < -3, 0 < x < 1
Example 6
 x2  3x  4  0
no real solution
Example 7
x2  2x  5  0
–5 < x < -4, 1 < x < 2
Example 8
x2  4x  5  0
-1, 5
Example 9
2x2  5  10x
0 < x < 1, 4 < x < 5
9-A4 Handout A4.
All graphs must be
completed on graph
paper – check out
the Colina website to
download coordinate
planes.