Chapter 1 Section 2: Quadratic Equations

Chapter 1 Section 2: Quadratic Equations • In this section, we will… • Solve quadratic equations by factoring • Solve quadratic equations by extraction of roots • Solve quadratic equations by completing the square • Solve quadratic equations by using the quadratic formula • (real solutions only) • Use the discriminant to determine the nature of the • real solutions of a given quadratic equation • Solve applications involving quadratic equations

quadratic equation • We will solve quadratic equations by: • factoring • extraction of roots • completing the square • using the quadratic formula • possible outcomes: standard form 1.2 Quadratic Equations

Solving Quadratic Equations by Factoring: 1. Write the quadratic equation in standard form (i.e. set the equation equal to 0) 2. Factor completely 3. Use the Zero-Product Rule 4. Solve the resulting linear equations 5. Check your potential solution(s) We will solve by factoring when, once set equal to zero, the result is factorable. 1.2 Quadratic Equations: Solving Quadratic Equations by Factoring

Examples: Solve each equation by factoring. Check your result(s). checks check 1.2 Quadratic Equations: Solving Quadratic Equations by Facoring

Example: Solve the equation by factoring. Check your result(s). checks 1.2 Quadratic Equations: Solving Quadratic Equations by Factoring

Solving Quadratic Equations by Extracting Roots: 1. Write the quadratic equation in the form 2. Take the square root of both sides of the equation • If c < 0, there will be no real solutions • If c = 0, there will be one real solution • If c > 0, there will be two real solutions 3. Check your potential solution(s) We will solve by extracting roots when our equation has the form 1.2 Quadratic Equations: Solving Quadratic Equations by Extraction of Roots

Examples: Solve each equation by extracting the roots (a.k.a. the square root method). Check your result(s). checks checks 1.2 Quadratic Equations: Solving Quadratic Equations by Extraction of Roots

Completing the Square: Recall our perfect squares from Review Section 4 example: We will be reversing this process and filling in the blanks. examples: Take half of the x-term coefficient It goes here Now square that result It goes here 1.2 Quadratic Equations: Solving Quadratic Equations by Completing the Square

Solving Quadratic Equations by Completing the Square: 1. Write the quadratic equation in the form 2. Make sure a = 1… if it is not, divide all terms by a 3. Complete the square • Find half of b • Square the result 4. Solve the resulting equation by extraction of roots 5. Check your potential solution(s) We can solve any quadratic eq. this way! 1.2 Quadratic Equations: Solving Quadratic Equations by Completing the Square

Examples: Solve the equation by completing the square. Check your result(s). checks 1.2 Quadratic Equations: Solving Quadratic Equations by Completing the Square

Solving Quadratic Equations by Using the Quadratic Formula: 1. Write the quadratic equation in the form 2. Use the quadratic formula to find the solution(s) 3. Check your potential solution(s) We can solve any quadratic eq. this way! • We can use the discriminant to determine the nature of the real solutions to our given quadratic equation. • If the discriminant of is: • negative, then there are no real solutions • zero, then there is one real solution • positive, then there are two different real solutions 1.2 Quadratic Equations: Solving Quad. Equations by Using the Quadratic Formula

Examples: Solve the equation by using the quadratic formula: Check your result(s). 1.2 Quadratic Equations: Solving Quad. Equations by Using the Quadratic Formula

Examples: Use the discriminant to determine the nature of the real solutions of the following quadratic equations. 1.2 Quadratic Equations: Using the Discriminant to Determine the Nature of Solutions

Summary of Techniques: We can now solve quadratic equations by: • Factoring • place in form • factor and use zero-product rule • Extracting the roots • place in form • take the square root of both sides • Completing the square • place in form • complete the square • take the square root of both sides • Using the quadratic formula • place in the form • use the quadratic formula Can only solve by factoring if this is factorable Can only extract roots if there is no x-term Take half of the x-term coefficient It goes here Can use to solve any quad. eq. Now square that result It goes here 1.2 Quadratic Equations

How to Solve a Word Problem: • Step 1: Read the problem until you understand it. • What are we asked to find? • What information is given? • What vocabulary is being used? • Step 2: Assign a variable to represent what you are looking for. • Express any remaining unknown quantities in terms of this variable. • Step 3: Make a list of all known facts and form an equation or inequality to solve. • It may help to make a labeled: diagram, table or chart, graph • Step 4: Solve • Step 5: State the solution in a complete sentence by mirroring the original question. • Be sure to include units when necessary. • Step 6: Check your result(s) in the words of the problem • Does your solution make sense? 1.2 Quadratic Equations: Solving Applications Involving Quadratic Equations

Example: The median weekly earnings E, in dollars, for full-time women workers ages 16 years and older from 2000 through 2008 can be estimated by the equation where x is the number of years after 2000. In what year will the median weekly earnings be $632. 1.2 Quadratic Equations: Solving Applications Involving Quadratic Equations

Example: The area of a rectangular window is to be 143 square feet. If the length is to be two feet more than the width, what are the dimensions? 1.2 Quadratic Equations: Solving Applications Involving Quadratic Equations

Example: A ball is thrown upward with an initial velocity of 20 meters per second. The distance s, in meters, of the object from the ground after t seconds is • When will the object be 15 meters above the ground? • When will it strike the ground? 1.2 Quadratic Equations: Solving Applications Involving Quadratic Equations

Independent Practice You learn math by doing math. The best way to learn math is to practice, practice, practice. The assigned homework examples provide you with an opportunity to practice. Be sure to complete every assigned problem (or more if you need additional practice). Check your answers to the odd-numbered problems in the back of the text to see whether you have correctly solved each problem; rework all problems that are incorrect. Read pp. 97-106 Homework: pp. 106-109 #15-23 odds, 29-37 odds, 41-45 odds, 53-59 odds, 71, 93-101 odds, 107 1.2 Quadratic Equations

Chapter 1 Section 2: Quadratic Equations