The Quadratic Equation Attendance Problems.II+2_6... · The Quadratic Equation Attendance Problems. ... These two zeros are the same distance, ! , ... 2t + 400. To the

Algebra II 2-6 Page � of �1 13

The Quadratic Equation Attendance Problems. Write each function in standard form.

1. ! 2. ! """"""""Evaluate b2 – 4ac for the given values of the valuables.

3. a = 2, b = 7, c = 5 4. a = 1, b = 3, c = -3 """""• I can solve quadratic equations using the Quadratic Formula. • I can classify roots using the discriminant. ""CCSS.MATH.CONTENT.HSN.CN.C.7 Solve quadratic equations with real coefficients that have complex solutions. CCSS.MATH.CONTENT.HSA.CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. "Vocabulary: Discriminant ""

f (x) = x − 4( )2 + 3 g(x) = 2 x + 6( )2 −11


5. Given ! , solve for x. """"""""""

"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.

ax2 + bx + c = 0

x = − b2a

± b2 − 4ac2a

b2 − 4ac2a

x = − b2a

To subtract fractions, you need a common denominator.

Remember!


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.

""


1E X A M P L E Quadratic Functions with Real Zeros

Find the zeros of f (x) = x 2 + 10x + 2 by using the Quadratic Formula.

x 2 + 10x + 2 = 0 Set f (x) = 0.

x = -b ± √ """" b 2 - 4ac __ 2a

Write the Quadratic Formula.

x = -10 ± √ """""" (10 ) 2 - 4 (1) (2)

___ 2 (1)

Substitute 1 for a, 10 for b, and 2 for c.

x = -10 ± √ """ 100 - 8 __ 2

= -10 ± √ " 92 __ 2

Simplify.

x = -10 ± 2 √ " 23 __ 2

= -5 ± √ " 23 Write in simplest form.

Check Solve by completing the square.

x 2 + 10x + 2 = 0

x 2 + 10x = -2

x 2 + 10x + 25 = -2 + 25

(x + 5) 2 = 23

x = -5 ± √ " 23 ✔

Find the zeros of each function by using the Quadratic Formula. 1a. f (x) = x 2 + 3x - 7 1b. g (x) = x 2 - 8x + 10

2E X A M P L E Quadratic Functions with Complex Zeros

Find the zeros of f (x) = 2 x 2 - x + 2 by using the Quadratic Formula.

2 x 2 - x + 2 = 0 Set f (x) = 0.

x = -b ± √ """" b 2 - 4ac __ 2a


x = - (-1) ± √ """"""" (-1) 2 - 4 (2) (2)

___ 2 (2)

Substitute 2 for a, -1 for b, and 2 for c.

x = 1 ± √ """ 1 - 16 __ 4

= 1 ± √ "" -15 _ 4

Simplify.

x = 1 ± i √ " 15 _ 4

= 1 _ 4

± √ " 15 _

4 i Write in terms of i.

2. Find the zeros of g (x) = 3 x 2 - x + 8 by using the Quadratic Formula.

The discriminant is part of the Quadratic Formula that you can use to determine the number of real roots of a quadratic equation.

x = -b ± √ """" b 2 - 4ac __ 2a

Discriminant

2-6 The Quadratic Formula 101

CS10_A2_MESE612263_C02L06.indd 101CS10_A2_MESE612263_C02L06.indd 101 2/24/11 4:41:58 PM2/24/11 4:41:58 PM


Example 1. Find the zeros of ! by using the quadratic formula. """""""""""""Guided Practice. Find the zeros of each function by using the quadratic formula.

6. ! 7. ! """""""""""

f x( ) = 2x2 −16x + 27

f x( ) = x2 + 3x − 7 g x( ) = x2 − 8x +10


Example 2. Find the zeros of ! by using the quadratic formula. """""""""""""

f x( ) = 4x2 + 3x + 2


1E X A M P L E Quadratic Functions with Real Zeros

Find the zeros of f (x) = x 2 + 10x + 2 by using the Quadratic Formula.

x 2 + 10x + 2 = 0 Set f (x) = 0.

x = -b ± √ """" b 2 - 4ac __ 2a


x = -10 ± √ """""" (10 ) 2 - 4 (1) (2)

___ 2 (1)

Substitute 1 for a, 10 for b, and 2 for c.

x = -10 ± √ """ 100 - 8 __ 2

= -10 ± √ " 92 __ 2

Simplify.

x = -10 ± 2 √ " 23 __ 2

= -5 ± √ " 23 Write in simplest form.

Check Solve by completing the square.

x 2 + 10x + 2 = 0

x 2 + 10x = -2

x 2 + 10x + 25 = -2 + 25

(x + 5) 2 = 23

x = -5 ± √ " 23 ✔

Find the zeros of each function by using the Quadratic Formula. 1a. f (x) = x 2 + 3x - 7 1b. g (x) = x 2 - 8x + 10

2E X A M P L E Quadratic Functions with Complex Zeros

Find the zeros of f (x) = 2 x 2 - x + 2 by using the Quadratic Formula.

2 x 2 - x + 2 = 0 Set f (x) = 0.

x = -b ± √ """" b 2 - 4ac __ 2a


x = - (-1) ± √ """"""" (-1) 2 - 4 (2) (2)

___ 2 (2)

Substitute 2 for a, -1 for b, and 2 for c.

x = 1 ± √ """ 1 - 16 __ 4

= 1 ± √ "" -15 _ 4

Simplify.

x = 1 ± i √ " 15 _ 4

= 1 _ 4

± √ " 15 _

4 i Write in terms of i.

2. Find the zeros of g (x) = 3 x 2 - x + 8 by using the Quadratic Formula.

The discriminant is part of the Quadratic Formula that you can use to determine the number of real roots of a quadratic equation.

x = -b ± √ """" b 2 - 4ac __ 2a

Discriminant




8. Guided Practice. Find the zeros of ! by using the quadratic formula. """""""""""""The discriminant is part of the Quadratic Formula that you can use to determine the number of real roots of a quadratic equation.

f x( ) = 3x2 − x + 8

Make sure the equation is in standard form before you evaluate the discriminant, b2 – 4ac.

Caution!


Example 3. Find the type and number of solutions for the equation. A. ! B. ! C. ! """"""""

Guided Practice. Find the type and number of solutions for the equation. 9. ! 10. ! 11. ! """""""""

x2 + 36 = 12x x2 + 40 = 12x x2 + 30 = 12x

x2 − 4x = −4 x2 − 4x = −8 x2 − 4x = 2

The discriminant of the quadratic equation a x 2 + bx + c = 0 (a ≠ 0) is b 2 - 4ac.

b 2 - 4ac > 0 b 2 - 4ac = 0 b 2 - 4ac < 0

two distinct real solutions one distinct real solution two distinct nonreal complex solutions

Discriminant

3E X A M P L E Analyzing Quadratic Equations by Using the Discriminant

Find the type and number of solutions for each equation.

A x 2 - 6x = -7 B x 2 - 6x = -9 C x 2 - 6x = -11

x 2 - 6x + 7 = 0 x 2 - 6x + 9 = 0 x 2 - 6x + 11 = 0

b 2 - 4ac b 2 - 4ac b 2 - 4ac

(-6) 2 - 4 (1) (7) (-6) 2 - 4 (1) (9) (-6) 2 - 4 (1) (11)

36 - 28 = 8 36 - 36 = 0 36 - 44 = -8

b 2 - 4ac > 0; b 2 - 4ac = 0; b 2 - 4ac < 0; the the equation the equation equation has two has two distinct has one distinct distinct nonreal real solutions. real solution. complex solutions.

Find the type and number of solutions for each equation. 3a. x 2 - 4x = -4 3b. x 2 - 4x = -8 3c. x 2 - 4x = 2

The graph shows the related functions for Example 3. Notice that the number of real solutions for the equation can be changed by changing the value of the constant c.

If I get integer roots when I use the Quadratic Formula, I know that I can quickly factor to check the roots. Look at my work for the equation x 2 - 7x + 10 = 0.

Double-Checking Roots

Quadratic Formula:

x = - (-7) ± √ ######## (-7) 2 - 4 (1) (10)

___ 2 (1)

= 7 ± √ # 9 _ 2 = 10 _

2 or 4 _

2 = 5 or 2

Factoring:

x 2 - 7x + 10 = 0 (x - 5) (x - 2) = 0 x = 5 or x = 2

Terry Cannon,Carver High School

6

-2

4

2

0 x

g(x) = x2 - 6x + 9

f(x) = x2 - 6x + 7

h(x) = x2 - 6x + 11

Make sure the equation is in standard form before you evaluate the discriminant, b 2 - 4ac.

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102 Chapter 2 Quadratic Functions



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. ""


Release point

Path ofwater

Target

x ft

4E X A M P L E Aviation Application

The 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 = -16 t 2 - 2t + 500. The horizontal distance x in feet between the water and its point of release is modeled by x = 91t. At what horizontal distance from the fire should the pilot start releasing the water in order to hit the target?

Step 1 Use the first 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 use the quadratic formula to solve for t.

y = -16 t 2 - 2t + 500

0 = -16 t 2 - 2t + 500 Set y equal to 0.

t = -b ± √ """" b 2 - 4ac __ 2a

Use the Quadratic Formula.

t = - (-2) ± √ """"""""" (-2) 2 - 4 (-16) (500)

___ 2 (-16)

Substitute for a, b, and c.

t = 2 ± √ """ 32,004

__ -32

Simplify.

t ≈ -5.65 or t ≈ 5.53

The time cannot be negative, so the water lands on the target about 5.5 seconds after it is released.

Step 2 Find the horizontal distance that the water will have traveled in this time.

x = 91t

x = 91 (5.5) Substitute 5.5 for t.

x = 500.5 Simplify.

The water will have traveled a horizontal distance of about 500 feet. Therefore, the pilot should start releasing the water when the horizontal distance between the helicopter and the fire is 500 feet.

Check Use substitution to check that the water hits the ground after about 5.53 seconds.

y = -16 t 2 - 2t + 500

y = -16 (5.53) 2 - 2 (5.53) + 500

y ≈ -0.3544 ✔ The height is approximately equal to 0 when t = 5.53.

Use the information given above to answer the following. 4. The pilot’s altitude decreases, which changes the function

describing the water’s height to y = -16 t 2 - 2t + 400. To the nearest foot, at what horizontal distance from the target should the pilot begin releasing the water?

Once you have found the value of t, you have solved only part of the problem. You will use this value to find the answer you are looking for.




Example 4. 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? """"""""""""""""""""""


12. Guided Practice. 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 pilot’s altitude decreases, which changes the function describing the water’s height to y = –16t2 –2t + 400. To the nearest foot, at what horizontal distance from the target should the pilot begin releasing the water? """""""""""""""


No matter which method you use to solve a quadratic equation, you should get the same answer.

Helpful Hint


"2-6 Assignment (p 105) 18, 21, 22, 24-26, 29, 36, 39, 40, 45-47, 52, 61-64. """""

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The Quadratic Equation Attendance Problems.II+2_6... · The Quadratic Equation Attendance Problems. ... These two zeros are the same distance, ! , ... 2t + 400. To the