College Algebra Twelfth Edition Chapter 1 Equations and
College Algebra Twelfth Edition Chapter 1 Equations and Inequalities Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 1
Section 1. 4 Quadratic Equations • The Zero-Factor Property • The Square Root Property • Completing the Square • The Quadratic Formula • Solving for a Specified Variable • The Discriminant Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 2
Quadratic Equation in One Variable An equation that can be written in the form where a, b, and c are real numbers with is a quadratic equation. The given form is called standard form. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 3
Second-degree Equation A quadratic equation is a second-degree equation—that is, an equation with a squared variable term and no terms of greater degree. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 4
Zero-Factor Property If a and b are complex numbers with then a = 0 or both equal zero. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 5
Example 1: Using the Zero-Factor Property Solve Solution Standard form Factor. Zero-factor property Solve each equation. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 6
Square Root Property (1 of 3) A quadratic equation of the form also be solved by factoring. can Subtract k. Factor. Zero-factor property. Solve each equation. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 7
Square Root Property (2 of 3) If then Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 8
Square Root Property (3 of 3) is That is, the solution set of which may be abbreviated Both solutions are real if k > 0, and both are pure imaginary if k < 0. If k < 0, we write the solution set as If k = 0, then there is only one distinct solution, 0, sometimes called a double solution. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 9
Example 2: Using the Square Root Property (1 of 3) Solve each quadratic equation. (a) Solution By the square root property, the solution set of Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 10
Example 2: Using the Square Root Property (2 of 3) Solve each quadratic equation. (b) Solution Since the solution set of is Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 11
Example 2: Using the Square Root Property (3 of 3) Solve each quadratic equation. (c) Solution Generalized square root property Add 4. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 12
Solving a Quadratic Equation by Completing the Square To solve where square, use these steps. by completing the divide both sides of the equation by a. Step 1 if Step 2 Rewrite the equation so that the constant term is alone on one side of the equality symbol. Step 3 Square half the coefficient of x, and add this square to each side of the equation. Step 4 Factor the resulting trinomial as a perfect square and combine like terms on the other side. Step 5 Use the square root property to complete the solution. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 13
Example 3: Using Completing the Square (a = 1) (1 of 2) Solve Solution Step 1 This step is not necessary since a = 1. Add 14 to each side. Step 2 Step 3 add 4 to each side. Step 4 Factor. Combine like terms. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 14
Example 3: Using Completing the Square (a = 1) (2 of 2) Solve Solution Step 5 Square root property. Add 2 to each side. Simplify the radical. The solution set is Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 15
Example 4: Using Completing the Square (a Not Equals 1) (1 of 3) Solve Solution Divide by 9. (Step 1) Subtract 1 from each side. (Step 2) to each side. (Step 3) Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 16
Example 4: Using Completing the Square (a Not Equals 1) (2 of 3) Solve Solution Factor. Combine like terms. (Step 4) Square root property (Step 5) Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 17
Example 3: Using Completing the Square (a Not Equals 1) (3 of 3) Solve square. by completing the Solution The solution set is Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 18
The Quadratic Formula If we start with the equation for a > 0, and complete the square to solve for x in terms of the constants a, b, and c, the result is a general formula for solving any quadratic equation. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 19
Quadratic Formula The solutions of the quadratic equation are given by the where quadratic formula. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 20
Caution (1 of 2) Remember to extend the fraction bar in the quadratic formula extends under the −b term in the numerator. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 21
Example 5: Using the Quadratic Formula (Real Solution) (1 of 2) Solve Solution Write in standard form. Here a = 1, b = − 4, c = 2. Quadratic formula Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 22
Example 5: Using the Quadratic Formula (Real Solution) (2 of 2) Solve Solution Simplify. Factor out 2 in the numerator. Lowest terms The solution set is Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 23
Example 6: Using the Quadratic Formula (Nonreal Complex Solutions) (1 of 2) Solve Solution Write in standard form. Quadratic formula with a = 2, b = − 1, c=4 Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 24
Example 6: Using the Quadratic Formula (Nonreal Complex Solutions) (2 of 2) Solve Solution Simplify. The solution set is Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 25
Example 7: Solving for a Quadratic Variable in a Formula (1 of 5) Solve each equation for the specified variable. Use when taking square roots. (a) Solution Multiply each side by 4. Divide each side by Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 26
Example 7: Solving for a Quadratic Variable in a Formula (2 of 5) Solve each equation for the specified variable. Use when taking square roots. (a) Solution Square root property Multiply by Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 27
Example 7: Solving for a Quadratic Variable in a Formula (3 of 5) Solve each equation for the specified variable. Use when taking square roots. (a) Solution Multiply numerators. Multiply denominators. Simplify the radical. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 28
Example 7: Solving for a Quadratic Variable in a Formula (4 of 5) Solve each equation for the specified variable. Use when taking square roots. (b) Solution Because has terms with use the quadratic formula. and t, Write in standard form. Quadratic formula Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 29
Example 7: Solving for a Quadratic Variable in a Formula (5 of 5) Solve each equation for the specified variable. Use when taking square roots. (b) Solution Here, a = r, b = −s, and c = −k. Simplify. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 30
Solving for a Specified Variable Note In Example 8, we took both positive and negative square roots. However, if the variable represents time or length in an application, we consider only the positive square root. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 31
The Discriminant (1 of 2) The Discriminant The quantity under the radical in the quadratic formula, is called the discriminant. Discriminant Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 32
The Discriminant (2 of 2) Discriminant Number of Solutions Type of Solutions Positive, perfect square Two Rational Positive, but not a perfect square Two Irrational One (a double solution) Rational Two Nonreal complex Zero Negative Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 33
Caution (2 of 2) The restriction on a, b, and c is important. For example, has discriminant which would indicate two rational solutions if the coefficients were integers. By the quadratic formula, the two solutions are irrational numbers. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 34
Example 8: Using the Discriminant (1 of 3) Evaluate the discriminant for each equation. Then use it to determine the number of distinct solutions, and tell whether they are rational, irrational, or nonreal complex numbers. (a) Solution The discriminant 84 is positive and not a perfect square, so there are two distinct irrational solutions. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 35
Example 8: Using the Discriminant (2 of 3) Evaluate the discriminant for each equation. Then use it to determine the number of distinct solutions, and tell whether they are rational, irrational, or nonreal complex numbers. (b) Solution First, write the equation in standard form as There is one distinct rational solution, a double solution. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 36
Example 8: Using the Discriminant (3 of 3) Evaluate the discriminant for each equation. Then use it to determine the number of distinct solutions, and tell whether they are rational, irrational, or nonreal complex numbers. (c) Solution There are two distinct nonreal complex solutions. (They are complex conjugates. ) Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 37
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