Holt McDougal Algebra 2 5-6 The Quadratic Formula 5-6 The Quadratic Formula Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson

Holt McDougal Algebra 2

5-6 The Quadratic Formula5-6 The Quadratic Formula

Holt Algebra 2

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5-6 The Quadratic Formula

Opener-Same Sheet-9/29Write each function in standard form.

Evaluate b2 – 4ac for the given values of the valuables.

1. 19 = x2 – 8x 2. -61 -24x = 2x2

f(x) = x2 – 8x + 19 g(x) = 2x2 + 24x + 61

4. a = 1, b = 3, c = –33. a = 2, b = 7, c = 5

9 21



5-5 Homework Quiz

Solve each equation.

2. 3x2 + 96 = 0

1. Express in terms of i.



Solve quadratic equations using the Quadratic Formula.

Classify roots using the discriminant.

Objectives



discriminant

Vocabulary



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.





To subtract fractions, you need a common denominator.

Remember!



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.



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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Find the zeros of f(x)= 2x2 – 16x + 27 using the Quadratic Formula.

Example 1: Quadratic Functions with Real Zeros

Write in simplest form.



Find the zeros of f(x) = x2 + 3x – 7 using the Quadratic Formula.

Check It Out! Example 1a

x2 + 3x – 7 = 0 x2 – 8x + 10 = 0



Find the zeros of f(x) = 4x2 + 3x + 2 using the Quadratic Formula.

Example 2: Quadratic Functions with Complex Zeros

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


5-6 The Quadratic FormulaOpener-SAME SHEET-9/29

1. Factor-Need H grapha. 2x2 + 9x + 10 b. 5x2 + 31x + 6



Reteach



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



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

Caution!



Explore



Opener-SAME SHEET-9/30

Solve by completing the square.

t2 – 8t – 5 = 0



Opener-SAME SHEET-9/30

Find the type and member of solutions for each equation.

1. x2 – 14x + 50 2. x2 – 14x + 48

2 distinct nonreal complex

2 distinct real



f(x)=x2 -12x + 36 f(x)=x2 -5x - 6



Opener-SAME SHEET-10/3• Radicals

1.Simpilfy

a. √-36 b. √-80 c. 4 √ -12

2. Solve

a. 4x2 + 32= 0













5-6 Homework Quiz AFind the zeros of each function by using the Quadratic Formula.

1. f(x) = x2 – 4x – 6



Find the zeros of f(x) = 4x2 + 3x + 2 using the Quadratic Formula.

Example 2: Quadratic Functions with Complex Zeros

f(x)=x2 -12x + 36



Find the type and number of solutions for the equation.

Example 3A: Analyzing Quadratic Equations by Using the Discriminant

x2 + 36 = 12x

one distinct real solution.

x2 + 40 = 12x

two distinct nonreal complex solutions.



Find the type and number of solutions for the equation.

Example 3C: Analyzing Quadratic Equations by Using the Discriminant

x2 + 30 = 12x

two distinct real solutions.

x2 – 4x = –4

one distinct real solution.



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.



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?

Example 4: Sports Application



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

x = 29.3t

Example 4 Continued

Substitute 1.77 for t.

Simplify.

x ≈ 29.3(1.77)

x ≈ 51.86

x ≈ 52

The shot put will have traveled a horizontal distance of about 52 feet.


5-6 The Quadratic FormulaProperties of Solving Quadratic Equations


5-6 The Quadratic FormulaProperties of Solving Quadratic Equations



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

Helpful Hint



Lesson Quiz: Part IFind 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 nonreal complex

2 distinct real



Lesson Quiz: Part II5. A pebble is tossed from the top of a cliff. The

pebble’s 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

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Holt McDougal Algebra 2 5-6 The Quadratic Formula 5-6 The Quadratic Formula Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson