Holt Algebra 2

5-6 The Quadratic Formula5-6 The Quadratic Formula

Holt Algebra 2

Warm UpLesson PresentationLesson Quiz

Holt Algebra 2

5-6 The Quadratic Formula

Warm Up

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

4. a = 1, b = 3, c = –33. a = 2, b = 7, c = 59 21

2. Solve the equation.3x2 + 96 = 0

1. Express in terms of i.

Holt Algebra 2

5-6 The Quadratic Formula

Solve quadratic equations using the Quadratic Formula.Classify roots using the discriminant.

Objectives

Holt Algebra 2

5-6 The Quadratic Formula

discriminantVocabulary

Holt Algebra 2

5-6 The Quadratic Formula

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.

Holt Algebra 2

5-6 The Quadratic Formula

Holt Algebra 2

5-6 The Quadratic Formula

To subtract fractions, you need a common denominator.

Remember!

Holt Algebra 2

5-6 The Quadratic Formula

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.

Holt Algebra 2

5-6 The Quadratic Formula

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.

Holt Algebra 2

5-6 The Quadratic Formula

Holt Algebra 2

5-6 The Quadratic Formula

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

Example 1: Quadratic Functions with Real Zeros

Set f(x) = 0.Write the Quadratic Formula.

Substitute 2 for a, –16 for b, and 27 for c.

Simplify.

Write in simplest form.

2x2 – 16x + 27 = 0

Holt Algebra 2

5-6 The Quadratic Formula

Check Solve by completing the square.

Example 1 Continued

Holt Algebra 2

5-6 The Quadratic Formula

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

Set f(x) = 0.

Write the Quadratic Formula.

Substitute 1 for a, 3 for b, and –7 for c.

Simplify.

Write in simplest form.

Check It Out! Example 1a

x2 + 3x – 7 = 0

Holt Algebra 2

5-6 The Quadratic Formula

Check Solve by completing the square.x2 + 3x – 7 = 0

x2 + 3x = 7

Check It Out! Example 1a Continued

Holt Algebra 2

5-6 The Quadratic Formula

Find the zeros of f(x)= x2 – 8x + 10 using the Quadratic Formula.

Set f(x) = 0.

Write the Quadratic Formula.

Substitute 1 for a, –8 for b, and 10 for c.

Simplify.

Write in simplest form.

Check It Out! Example 1b

x2 – 8x + 10 = 0

Holt Algebra 2

5-6 The Quadratic Formula

Check Solve by completing the square.x2 – 8x + 10 = 0

x2 – 8x = –10 x2 – 8x + 16 = –10 + 16

(x + 4)2 = 6

Check It Out! Example 1b Continued

Holt Algebra 2

5-6 The Quadratic Formula

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

Example 2: Quadratic Functions with Complex Zeros

Set f(x) = 0.

Write the Quadratic Formula.

Substitute 4 for a, 3 for b, and 2 for c.

Simplify.

Write in terms of i.

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

Holt Algebra 2

5-6 The Quadratic Formula

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.

Check It Out! Example 2

Holt Algebra 2

5-6 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.

Holt Algebra 2

5-6 The Quadratic Formula

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

Caution!

Holt Algebra 2

5-6 The Quadratic Formula

Find the type and number of solutions for the equation.

Example 3A: Analyzing Quadratic Equations by Using the Discriminant

x2 + 36 = 12xx2 – 12x + 36 = 0b2 – 4ac(–12)2 – 4(1)(36)144 – 144 = 0

b2 – 4ac = 0

The equation has one distinct real solution.

Holt Algebra 2

5-6 The Quadratic Formula

Find the type and number of solutions for the equation.

Example 3B: Analyzing Quadratic Equations by Using the Discriminant

x2 + 40 = 12xx2 – 12x + 40 = 0b2 – 4ac(–12)2 – 4(1)(40)144 – 160 = –16

b2 –4ac < 0

The equation has two distinct nonreal complex solutions.

Holt Algebra 2

5-6 The Quadratic Formula

Find the type and number of solutions for the equation.

Example 3C: Analyzing Quadratic Equations by Using the Discriminant

x2 + 30 = 12xx2 – 12x + 30 = 0b2 – 4ac(–12)2 – 4(1)(30)144 – 120 = 24

b2 – 4ac > 0

The equation has two distinct real solutions.

Holt Algebra 2

5-6 The Quadratic FormulaCheck It Out! Example 3a

Find the type and number of solutions for the equation.

x2 – 4x = –4x2 – 4x + 4 = 0b2 – 4ac(–4)2 – 4(1)(4)16 – 16 = 0

b2 – 4ac = 0

The equation has one distinct real solution.

Holt Algebra 2

5-6 The Quadratic FormulaCheck It Out! Example 3b

Find the type and number of solutions for the equation.

x2 – 4x = –8x2 – 4x + 8 = 0b2 – 4ac(–4)2 – 4(1)(8)16 – 32 = –16

b2 – 4ac < 0

The equation has two distinct nonreal complex solutions.

Holt Algebra 2

5-6 The Quadratic FormulaCheck It Out! Example 3c

Find the type and number of solutions for each equation.

x2 – 4x = 2x2 – 4x – 2 = 0b2 – 4ac(–4)2 – 4(1)(–2)16 + 8 = 24

b2 – 4ac > 0

The equation has two distinct real solutions.

Holt Algebra 2

5-6 The Quadratic FormulaProperties of Solving Quadratic Equations

Holt Algebra 2

5-6 The Quadratic FormulaProperties of Solving Quadratic Equations

Holt Algebra 2

5-6 The Quadratic Formula

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

Helpful Hint

Holt Algebra 2

5-6 The Quadratic FormulaLesson 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 + 482 distinct nonreal complex

2 distinct real