The Quadratic Formula

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The Quadratic Formula
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2
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Warm Up Write each function in standard
form. Evaluate b2 4ac for the given values
of the valuables.
2. g(x) 2(x 6)2 11
1. f(x) (x 4)2 3
g(x) 2x2 24x 61
f(x) x2 8x 19
4. a 1, b 3, c 3
3. a 2, b 7, c 5
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Objectives
Solve quadratic equations using the Quadratic
Formula. Classify roots using the discriminant.
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Vocabulary
discriminant
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You have learned several methods for solving
quadratic equations graphing, making tables,
factoring, using square roots, and completing the
square. Another method is to use the Quadratic
Formula, which allows you to solve a quadratic
equation in standard form.
By completing the square on the standard form of
a quadratic equation, you can determine the
Quadratic Formula.
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The symmetry of a quadratic function is evident
in the last step, . These two
zeros are the same distance, , away
from the axis of symmetry, ,with one zero
on either side of the vertex.
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You can use the Quadratic Formula to solve any
quadratic equation that is written in standard
form, including equations with real solutions or
complex solutions.
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Example 1 Quadratic Functions with Real Zeros
Find the zeros of f(x) 2x2 16x 27 using the
Quadratic Formula.
Set f(x) 0.
2x2 16x 27 0
Write the Quadratic Formula.
Substitute 2 for a, 16 for b, and 27 for c.
Simplify.
Write in simplest form.
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Example 1 Continued
Check Solve by completing the square.
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Check It Out! Example 1a
Find the zeros of f(x) x2 3x 7 using the
Quadratic Formula.
x2 3x 7 0
Set f(x) 0.
Write the Quadratic Formula.
Substitute 1 for a, 3 for b, and 7 for c.
Simplify.
Write in simplest form.
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Check It Out! Example 1a Continued
Check Solve by completing the square.
x2 3x 7 0
x2 3x 7
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Check It Out! Example 1b
Find the zeros of f(x) x2 8x 10 using the
Quadratic Formula.
x2 8x 10 0
Set f(x) 0.
Write the Quadratic Formula.
Substitute 1 for a, 8 for b, and 10 for c.
Simplify.
Write in simplest form.
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Check It Out! Example 1b Continued
Check Solve by completing the square.
x2 8x 10 0
x2 8x 10
x2 8x 16 10 16
(x 4)2 6
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Example 2 Quadratic Functions with Complex Zeros
Find the zeros of f(x) 4x2 3x 2 using the
Quadratic Formula.
Set f(x) 0.
f(x) 4x2 3x 2
Write the Quadratic Formula.
Substitute 4 for a, 3 for b, and 2 for c.
Simplify.
Write in terms of i.
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Check It Out! Example 2
Find the zeros of g(x) 3x2 x 8 using the
Quadratic Formula.
Set f(x) 0
Write the Quadratic Formula.
Substitute 3 for a, 1 for b, and 8 for c.
Simplify.
Write in terms of i.
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The discriminant is part of the Quadratic Formula
that you can use to determine the number of real
roots of a quadratic equation.
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Example 3A Analyzing Quadratic Equations by
Using the Discriminant
Find the type and number of solutions for the
equation.
x2 36 12x
x2 12x 36 0
b2 4ac
(12)2 4(1)(36)
144 144 0
b2 4ac 0
The equation has one distinct real solution.
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Example 3B Analyzing Quadratic Equations by
Using the Discriminant
Find the type and number of solutions for the
equation.
x2 40 12x
x2 12x 40 0
b2 4ac
(12)2 4(1)(40)
144 160 16
b2 4ac lt 0
The equation has two distinct nonreal complex
solutions.
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Example 3C Analyzing Quadratic Equations by
Using the Discriminant
Find the type and number of solutions for the
equation.
x2 30 12x
x2 12x 30 0
b2 4ac
(12)2 4(1)(30)
144 120 24
b2 4ac gt 0
The equation has two distinct real solutions.
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Check It Out! Example 3a
Find the type and number of solutions for the
equation.
x2 4x 4
x2 4x 4 0
b2 4ac
(4)2 4(1)(4)
16 16 0
b2 4ac 0
The equation has one distinct real solution.
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Check It Out! Example 3b
Find the type and number of solutions for the
equation.
x2 4x 8
x2 4x 8 0
b2 4ac
(4)2 4(1)(8)
16 32 16
b2 4ac lt 0
The equation has two distinct nonreal complex
solutions.
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Check It Out! Example 3c
Find the type and number of solutions for each
equation.
x2 4x 2
x2 4x 2 0
b2 4ac
(4)2 4(1)(2)
16 8 24
b2 4ac gt 0
The equation has two distinct real solutions.
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The graph shows related functions. Notice that
the number of real solutions for the equation can
be changed by changing the value of the constant
c.
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Example 4 Sports Application
An athlete on a track team throws a shot put. The
height y of the shot put in feet t seconds after
it is thrown is modeled by y 16t2 24.6t
6.5. The horizontal distance x in between the
athlete and the shot put is modeled by x 29.3t.
To the nearest foot, how far does the shot put
land from the athlete?
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Example 4 Continued
Step 1 Use the first equation to determine how
long it will take the shot put to hit the
ground. Set the height of the shot put
equal to 0 feet, and the use the quadratic
formula to solve for t.
y 16t2 24.6t 6.5
Set y equal to 0.
0 16t2 24.6t 6.5
Use the Quadratic Formula.
Substitute 16 for a, 24.6 for b, and 6.5 for c.
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Example 4 Continued
Simplify.
The time cannot be negative, so the shot put hits
the ground about 1.8 seconds after it is released.
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Example 4 Continued
Step 2 Find the horizontal distance that the
shot put will have traveled in this time.
x 29.3t
Substitute 1.77 for t.
x 29.3(1.77)
x 51.86
Simplify.
x 52
The shot put will have traveled a horizontal
distance of about 52 feet.
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Example 4 Continued
Check Use substitution to check that the shot
put hits the ground after about 1.77 seconds.
y 16t2 24.6t 6.5
y 16(1.77)2 24.6(1.77) 6.5
y 50.13 43.54 6.5
y 0.09
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The height is approximately equal to 0 when t
1.77.
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Check It Out! Example 4
A pilot of a helicopter plans to release a bucket
of water on a forest fire. The height y in feet
of the water t seconds after its release is
modeled by y 16t2 2t 500. the horizontal
distance x in feet between the water and its
point of release is modeled by x 91t.
The pilots altitude decreases, which changes the
function describing the waters height toy
16t2 2t 400. To the nearest foot, at what
horizontal distance from the target should the
pilot begin releasing the water?
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Check It Out! Example 4 Continued
Step 1 Use the equation to determine how long it
will take the water to hit the ground. Set the
height of the water equal to 0 feet, and then use
the quadratic formula for t.
y 16t2 2t 400
Set y equal to 0.
0 16t2 2t 400
Write the Quadratic Formula.
Substitute 16 for a, 2 for b, and 400 for c.
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Check It Out! Example 4
Simplify.
t 5.063 or t 4.937
The time cannot be negative, so the water lands
on a target about 4.937 seconds after it is
released.
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Check It Out! Example 4 Continued
Step 2 The horizontal distance x in feet between
the water and its point of release is
modeled by x 91t. Find the horizontal
distance that the water will have traveled
in this time.
x 91t
Substitute 4.937 for t.
x 91(4.937)
Simplify.
x 449.267
x 449
The water will have traveled a horizontal
distance of about 449 feet. Therefore, the pilot
should start releasing the water when the
horizontal distance between the helicopter and
the fire is 449 feet.
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Check It Out! Example 4 Continued
Check Use substitution to check that the water
hits the ground after about 4.937 seconds.
y 16t2 2t 400
y 16(4.937)2 2(4.937) 400
y 389.983 9.874 400
y 0.143
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The height is approximately equal to 0 when t
4.937.
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Properties of Solving Quadratic Equations
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Properties of Solving Quadratic Equations
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Lesson Quiz Part I
Find the zeros of each function by using the
Quadratic Formula.
1. f(x) 3x2 6x 5
2. g(x) 2x2 6x 5
Find the type and member of solutions for each
equation.
3. x2 14x 50
4. x2 14x 48
2 distinct real
2 distinct nonreal complex
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Lesson Quiz Part II
5. A pebble is tossed from the top of a cliff.
The pebbles height is given by y(t) 16t2
200, where t is the time in seconds. Its
horizontal distance in feet from the base of the
cliff is given by d(t) 5t. How far will the
pebble be from the base of the cliff when it hits
the ground?
about 18 ft