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Solving
Quadratic
Equations
Solving
Quadratic
Equations
9-5
9-5 byby
Graphing
Graphing
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Holt
McDougal
Algebra 1Algebra
Algebra11
Holt
McDougal
Solving Quadratic Equations
9-5 by Graphing
Warm Up
1. Graph y = x2 + 4x + 3.
2. Identify the vertex and zeros of the
function above.
vertex:(–2 , –1);
zeros:–3, –1
Holt McDougal Algebra 1
Solving Quadratic Equations
9-5 by Graphing
Objective
Solve quadratic equations by graphing.
Holt McDougal Algebra 1
Solving Quadratic Equations
9-5 by Graphing
Vocabulary
quadratic equation
Holt McDougal Algebra 1
Solving Quadratic Equations
9-5 by Graphing
Every quadratic function has a related quadratic
equation. A quadratic equation is an equation
that can be written in the standard form ax2 +
bx + c = 0, where a, b, and c are real numbers
and a ≠ 0.
When writing a quadratic function as its related
quadratic equation, you replace y with 0. So y = 0.
y = ax2 + bx + c
0 = ax2 + bx + c
ax2 + bx + c = 0
Holt McDougal Algebra 1
Solving Quadratic Equations
9-5 by Graphing
One way to solve a quadratic equation in standard
form is to graph the related function and find the
x-values where y = 0. In other words, find the
zeros of the related function. Recall that a
quadratic function may have two, one, or no zeros.
Holt McDougal Algebra 1
Solving Quadratic Equations
9-5 by Graphing
Example 1A: Solving Quadratic Equations by Graphing
Solve the equation by graphing the related
function.
2x2 – 18 = 0
Step 1 Write the related function.
2x2 – 18 = y, or y = 2x2 + 0x – 18
Step 2 Graph the function.
x=0
• The axis of symmetry is x = 0.
• The vertex is (0, –18).
• Two other points (2, –10) and
(3, 0)
• Graph the points and reflect them
across the axis of symmetry.
Holt McDougal Algebra 1
●
●
(3, 0)
●
●
(2, –10)
●
(0, –18)
Solving Quadratic Equations
9-5 by Graphing
Example 1A Continued
Solve the equation by graphing the related
function.
2x2 – 18 = 0
Step 3 Find the zeros.
The zeros appear to be 3 and –3.
Check 2x2 – 18 = 0
2(3)2 – 18
2(9) – 18
18 – 18
0
Holt McDougal Algebra 1
2x2 – 18 = 0
0
2(–3)2 – 18 0
0 Substitute 3 and –3 2(9) – 18 0
0 for x in the quadratic 18 – 18 0

0 equation.
Solving Quadratic Equations
9-5 by Graphing
Example 1B: Solving Quadratic Equations by Graphing
Solve the equation by graphing the related
function.
–12x + 18 = –2x2
Step 1 Write the related function.
y = –2x2 + 12x – 18
x=3
Step 2 Graph the function.
• The axis of symmetry is x = 3.
• The vertex is (3, 0).
• Two other points (5, –8) and
(4, –2).
• Graph the points and reflect them
across the axis of symmetry.
Holt McDougal Algebra 1
●
●
●
●
(3, 0)
● (4, –2)
● (5, –8)
Solving Quadratic Equations
9-5 by Graphing
Example 1B Continued
Solve the equation by graphing the related
function.
–12x + 18 = –2x2
Step 3 Find the zeros.
The only zero appears to be 3.
Check y = –2x2 + 12x – 18
0
–2(3)2 + 12(3) – 18
0
–18 + 36 – 18
0
0
You can also confirm the solution by using the Table
function. Enter the function and press
When y = 0, x = 3. The x-intercept is 3.
Holt McDougal Algebra 1
Solving Quadratic Equations
9-5 by Graphing
Example 1C: Solving Quadratic Equations by Graphing
Solve the equation by graphing the related
function.
2x2 + 4x = –3
Step 1 Write the related function.
2x2 + 4x + 3 = 0
y = 2x2 + 4x + 3
Step 2 Graph the function.
Use a graphing calculator.
Step 3 Find the zeros.
The function appears to
have no zeros.
Holt McDougal Algebra 1
Solving Quadratic Equations
9-5 by Graphing
Example 1C: Solving Quadratic Equations by Graphing
Solve the equation by graphing the related
function.
2x2 + 4x = –3
The equation has no real-number solutions.
Check reasonableness Use the table function.
There are no zeros in the Y1 column.
Also, the signs of the values in this
column do not change. The function
appears to have no zeros.
Holt McDougal Algebra 1
Solving Quadratic Equations
9-5 by Graphing
Check It Out! Example 1a
Solve the equation by graphing the related
function.
x2 – 8x – 16 = 2x2
Step 1 Write the related function.
y = x2 + 8x + 16
Step 2 Graph the function.
• The axis of symmetry is x = –4.
• The vertex is (–4, 0).
• The y-intercept is 16.
• Two other points are (–3, 1) and
(–2, 4).
• Graph the points and reflect them
across the axis of symmetry.
Holt McDougal Algebra 1
x = –4
●(–2 , 4)
●
●
●
● (–3, 1)
(–4, 0)
Solving Quadratic Equations
9-5 by Graphing
Check It Out! Example 1a Continued
Solve the equation by graphing the related
function.
x2 – 8x – 16 = 2x2
Step 3 Find the zeros.
The only zero appears to be –4.
Check y = x2 + 8x + 16
0
0
0
Holt McDougal Algebra 1
(–4)2 + 8(–4) + 16
16 – 32 + 16
0
Solving Quadratic Equations
9-5 by Graphing
Check It Out! Example 1b
Solve the equation by graphing the related
function.
6x + 10 = –x2
Step 1 Write the related function.
y = x2 + 6x + 10
Step 2 Graph the function.
• The axis of symmetry is x = –3 .
• The vertex is (–3 , 1).
• The y-intercept is 10.
• Two other points (–1, 5) and
(–2, 2)
• Graph the points and reflect them
across the axis of symmetry.
Holt McDougal Algebra 1
x = –3
● (–1, 5)
●
●
●
● (–2, 2)
(–3, 1)
Solving Quadratic Equations
9-5 by Graphing
Check It Out! Example 1b Continued
Solve the equation by graphing the related
function.
6x + 10 = –x2
Step 3 Find the zeros.
There appear to be no zeros.
You can confirm the solution
by using the Table function.
Enter the function and press
There are no negative
terms in the Y1 column.
Holt McDougal Algebra 1
Solving Quadratic Equations
9-5 by Graphing
Check It Out! Example 1c
Solve the equation by graphing the related
function.
–x2 + 4 = 0
Step 1 Write the related function.
y = –x2 + 4
Step 2 Graph the function.
Use a graphing calculator.
Step 3 Find the zeros.
The function appears to have
zeros at (2, 0) and (–2, 0).
Holt McDougal Algebra 1
Solving Quadratic Equations
9-5 by Graphing
Check It Out! Example 1c Continued
Solve the equation by graphing the related
function.
–x2 + 4 = 0
The equation has two real-number solutions.
Check reasonableness Use the table function.
There are two zeros in the Y1
column. The function appears to
have zeros at –2 and 2.
Holt McDougal Algebra 1
Solving Quadratic Equations
9-5 by Graphing
Example 2: Application
A frog jumps straight up from the ground.
The quadratic function f(t) = –16t2 + 12t
models the frog’s height above the ground
after t seconds. About how long is the frog
in the air?
When the frog leaves the ground, its height is
0, and when the frog lands, its height is 0. So
solve 0 = –16t2 + 12t to find the times when
the frog leaves the ground and lands.
Step 1 Write the related function
0 = –16t2 + 12t
y = –16t2 + 12t
Holt McDougal Algebra 1
Solving Quadratic Equations
9-5 by Graphing
Example 2 Continued
Step 2 Graph the function.
Use a graphing calculator.
Step 3 Use
to estimate the
zeros.
The zeros appear to be 0 and 0.75.
The frog leaves the ground at 0
seconds and lands at 0.75
seconds.
The frog is off the ground for
about 0.75 seconds.
Holt McDougal Algebra 1
Solving Quadratic Equations
9-5 by Graphing
Example 2 Continued
Check 0 =
0
0
0
0
–16t2 + 12t
–16(0.75)2 + 12(0.75)
–16(0.5625) + 9
–9 + 9
0
Holt McDougal Algebra 1
Substitute 0.75 for t
in the quadratic
equation.
Solving Quadratic Equations
9-5 by Graphing
Check It Out! Example 2
What if…? A dolphin jumps out of the water.
The quadratic function y = –16x2 + 32 x
models the dolphin’s height above the water
after x seconds. How long is the dolphin out of
the water?
When the dolphin leaves the water, its height is
0, and when the dolphin reenters the water, its
height is 0. So solve 0 = –16x2 + 32x to find
the times when the dolphin leaves and reenters
the water.
Step 1 Write the related function
0 = –16x2 + 32x
y = –16x2 + 32x
Holt McDougal Algebra 1
Solving Quadratic Equations
9-5 by Graphing
Check It Out! Example 2 Continued
Step 2 Graph the function.
Use a graphing calculator.
Step 3 Use
to estimate the
zeros.
The zeros appear to be 0 and 2.
The dolphin leaves the water
at 0 seconds and reenters at
2 seconds.
The dolphin is out of the
water for 2 seconds.
Holt McDougal Algebra 1
Solving Quadratic Equations
9-5 by Graphing
Check It Out! Example 2 Continued
Check 0 = –16x2 + 32x
0 –16(2)2 + 32(2)
0 –16(4) + 64
0
–64 + 64
0
0
Holt McDougal Algebra 1
Substitute 2 for x in
the quadratic
equation.
Solving Quadratic Equations
9-5 by Graphing
Lesson Quiz
Solve each equation by graphing the related
function.
1. 3x2 – 12 = 0 2, –2
2. x2 + 2x = 8 –4, 2
3. 3x – 5 = x2
no solution
4. 3x2 + 3 = 6x 1
5. A rocket is shot straight up from the ground.
The quadratic function f(t) = –16t2 + 96t
models the rocket’s height above the ground
after t seconds. How long does it take for the
rocket to return to the ground? 6 s
Holt McDougal Algebra 1