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**1 Equations and Inequalities Sections 1.1–1.4**

© 2008 Pearson Addison-Wesley. All rights reserved

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**Equations and Inequalities**

1 1.3 Complex Numbers 1.4 Quadratic Equations

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1.3 Complex Numbers Basic Concepts of Complex Numbers ▪ Operations on Complex Numbers Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

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**Basic Concepts of Complex Numbers**

The set of real numbers is a subset of the overall set of Complex Numbers. In order to extend the set of real numbers to include even roots of negative numbers such as , the number i is defined to have the following property: i2 = -1. From this definition, we have the imaginary unit, Numbers in the form a + bi where a and b are real numbers, are called complex numbers. In the complex Number a + bi , a is the real part and b is the imaginary part. Complex Numbers with a + bi Real Numbers with a + bi where b = 0 Rational Numbers Integers Non-Integers Irrational Numbers Imaginary Numbers with a + bi where a =0 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

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**Complex Numbers, cont’d**

The standard form of a complex number is written as a + bi , with the real and imaginary parts separated. Now when dealing with negative values under an even radical, the negative sign can be separated as follows, for a > 0: Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

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**1.3 Example 1 Writing √–a as i√a (page 104)**

Write as the product of a real number and i. (a) (b) (c) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

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**Multiply or divide. Simplify each answer.**

1.3 Example 2 Finding Products and Quotients Involving Negative Radicands (page 105) Multiply or divide. Simplify each answer. (a) (b) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

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**Multiply or divide. Simplify each answer.**

1.3 Example 2 Finding Products and Quotients Involving Negative Radicands (cont.) Multiply or divide. Simplify each answer. (c) (d) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

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**Write in standard form a + bi.**

1.3 Example 3 Simplifying a Quotient Involving a Negative Radicand (page 105) Write in standard form a + bi. Factor. Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

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**Operations with Complex Numbers**

Addition and Subtraction: Multiplication: Powers of i : Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

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Complex Conjugates For a complex number a + bi , its conjugate has the form, a - bi . The pair is referred to as complex conjugates. For real numbers a and b, Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

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**1.3 Example 4 Adding and Subtracting Complex Numbers (page 106)**

Find each sum or difference. (a) (4 – 5i) + (–5 + 8i) = [4 + (–5)] + (–5i + 8i) = –1 + 3i (b) (–6 + 3i) + (12 – 9i) = 6 – 6i (c) (–10 + 7i) – (5 – 3i) = (–10 – 5) + [7i + (3i)] = – i (d) (15 – 8i) – (–10 + 4i) = 25 – 12i Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

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**1.3 Example 5 Multiplying Complex Numbers (page 107)**

Find each product. (a) (5 + 3i)(2 – 7i) = 5(2) + (5)(–7i) + (3i)(2) + (3i)(–7i) = 10 – 35i + 6i – 21i2 = 10 – 29i – 21(–1) = 31 – 29i (b) (4 – 5i)2 = 42 – 2(4)(5i) + (5i)2 = 16 – 40i + 25i2 = 16 – 40i + 25(–1) = –9 – 40i Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

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**1.3 Example 5 Multiplying Complex Numbers (cont.)**

Find the product. (c) (9 – 8i)(9 + 8i) = 92 – (8i)2 = 81 – 64i2 = 81 – 64(–1) = = 145 or i This screen shows how the TI–83/84 Plus displays the results in this example. Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

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**1.3 Example 6 Simplifying Powers of i (page 107)**

Simplify each power of i. (a) (b) Write the given power as a product involving or (a) (b) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

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**1.3 Example 7(a) Dividing Complex Numbers (page 108)**

Write in standard form a + bi. Multiply the numerator and denominator by the complex conjugate of the denominator. Multiply. i2 = –1 Combine terms. Lowest terms; standard form Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

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**1.3 Example 7(b) Dividing Complex Numbers (page 108)**

Write in standard form a + bi. Multiply the numerator and denominator by the complex conjugate of the denominator. –i2 = 1 Multiply. Lowest terms; standard form This screen shows how the TI–83/84 Plus displays the results in this example. Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

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1.4 Quadratic Equations Solving a Quadratic Equation ▪ Completing the Square ▪ The Quadratic Formula ▪ Solving for a Specified Variable ▪ The Discriminant Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

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**Solving Quadratic Equations**

An equation that can be written in the form where a, b, and c are real numbers with a ≠ 0, is a quadratic equation. The given form is called the standard form. A quadratic equation is a second-degree equation – that is, an equation with a squared variable term as the highest degree. Zero-Factor Property If a and b are complex numbers with ab = 0, then a = 0 or b = 0 or both. Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

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**1.4 Example 1 Using the Zero-Factor Property (page 111)**

Solve Factor. or Set each factor equal to 0 and then solve for x. or or Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

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**1.4 Example 1 Using the Zero-Factor Property (cont.)**

Now check. Solution set: Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

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**Solving Quadratic Equations**

Square Root Property If x2 = k, then That is the solution set of the equation includes the postive and the negative values. If k = 0, then there is a double solution. Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

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**1.4 Example 2 Using the Square Root Property (page 112)**

Solve each quadratic equation. (a) (b) (c) Generalized square root property Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

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**Completing the Square To solve by completing the square:**

If a ≠ 1, divide both sides of the equation by a. Rewrite the equation so that the constant term is alone on one side of the equality symbol. Square half the coefficient of x, and add this square to both sides of the equation. Factor the resulting trinomial as a perfect square and combine the constant terms on the other side. Use the square root property to complete the solution. Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

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**Solve by completing the square.**

1.4 Example 3 Using the Method of Completing the Square, a = 1 (page 113) Solve by completing the square. Rewrite the equation so that the constant is alone on one side of the equation. Square half the coefficient of x, and add this square to both sides of the equation. Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

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**1.4 Example 3 Using the Method of Completing the Square, a = 1 (cont.)**

Factor the resulting trinomial as a perfect square and combine terms on the other side. Use the square root property to complete the solution. Solution set: Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

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**Solve by completing the square.**

1.4 Example 4 Using the Method of Completing the Square, a ≠ 1 (page 113) Solve by completing the square. Divide both sides of the equation by a, 4. Rewrite the equation so that the constant is alone on one side of the equation. Square half the coefficient of x, and add this square to both sides of the equation. Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

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**1.4 Example 4 Using the Method of Completing the Square, a ≠ 1 (cont.)**

Factor the resulting trinomial as a perfect square and combine terms on the other side. Use the square root property to complete the solution. Solution set: Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

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The Quadratic Formula The method of completing the square can be used to solve any quadratic equation. If this method is applied to the standard form for the quadratic equation, the results is the Quadratic Formula: Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

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**1.4 Example 5 Using the Quadratic Formula (Real Solutions) (page 115)**

Solve Write the equation in standard form. a = 1, b = 6, c = –3 Quadratic formula Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

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**1.4 Example 5 Using the Quadratic Formula (Real Solutions) (cont.)**

Solution set: Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

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**1.4 Example 6 Using the Quadratic Formula (Nonreal Complex Solutions) (page 115)**

Solve Write the equation in standard form. a = 4, b = –3, c = 5 Quadratic formula Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

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**1.4 Example 6 Using the Quadratic Formula (Nonreal Complex Solutions) (page 115)**

Solution set: Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

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**Solve for r. Use ± when taking square roots.**

1.4 Example 8(a) Solving for a Quadratic Variable in a Formula (page 116) Solve for r. Use ± when taking square roots. Goal: Isolate r. Multiply by 3. Divide by πh. Square root property Rationalize the denominator. Simplify. Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

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The Discriminant The discriminant is the quantity under the radical sign in the Quadratic Formula. The value of the discriminant is used to determine the number and type of solutions of a quadratic equation. Discriminant Number of Solutions Type of Solutions Positive, perfect square Two Rational Positive, but not a perfect square Irrational Zero One (double solution) Negative Nonreal complex Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

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**1.4 Example 9(a) Using the Discriminant (page 118)**

Determine the number of distinct solutions, and tell whether they are rational, irrational, or nonreal complex numbers. a = 4, b = –12, c = 9 There is one distinct rational solution. Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

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**1.4 Example 9(b) Using the Discriminant (page 118)**

Determine the number of distinct solutions, and tell whether they are rational, irrational, or nonreal complex numbers. Write in standard form. a = 3, b = 1, c = 5 There are two distinct nonreal complex solutions. Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

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**1.4 Example 9(c) Using the Discriminant (page 118)**

Determine the number of distinct solutions, and tell whether they are rational, irrational, or nonreal complex numbers. Write in standard form. a = 2, b = –6, c = –7 There are two distinct irrational solutions. Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

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