Solving Quadratic Equations by Graphs and Factoring

Transcription

1 Solving Quadratic Equations b Graphs and Factoring Algebra Unit: 05 Lesson: 0 Consider the equation = 0. Show (numericall) that = 5 is a solution. There is also another solution to the equation. Use the table to help find this second answer Solution? YES 6 Using the table above, plot the points (, ) to sketch the graph of = Which two points represent the solutions to the original equation? Write down their coordinates. Where are both of these points located? Here, we found both solutions to a quadratic equation using a method like trial-and-error. However, there are other (more efficient) methods for solving such quadratic equations: Part I: Graphs and Tables Part II: Factoring Part III: Quadratic Formula In these cases, we use the following terms to describe the solutions to quadratic equations = 0 When we are dealing with general equations, When we are dealing with functions, like this one, with s (but no s), such as = , then the answers are called then the answers are called or or Here, we could sa, Here, we could sa, The solutions to = 0 are = and = 4. The zeros of the function = occur at = and = 4. 01, TESCCC 09/06/1 page 1 of 6

2 Solving Quadratic Equations b Graphs and Factoring Algebra Unit: 05 Lesson: 0 Part I: Graphs and Tables Xmin= -9.4 Ymin= -1 WINDOW: Use a graphing calculator to complete the chart. Xma= 9.4 Yma= 1 Factored Equation Epanded Equation Graph (Sketch) Zeros of Function Check (with Table) E. = ( - )( + ) = ( )( + ) = + 4 = 4 = - = ) = ( - 4)( + 1) ) = ( + 3)( + 6) 3) = ( - 5)( - 5), or = ( - 5) 4) = - ( -.5)( + ) 5) = ) = , TESCCC 09/06/1 page of 6

3 Solving Quadratic Equations b Graphs and Factoring Questions: 7) How is each factored equation related to the zeros of its related function? Algebra Unit: 05 Lesson: 0 8) Equation #3 used the same factor twice: = ( - 5)( - 5). How does this affect the graph of its related function? 9) Equation #4 used a negative coefficient in front of its factors: = - ( -.5)( + ). How does this affect the graph of its related function? Think About It: The Fundamental Theorem of Algebra states that a polnomial equation of degree n has precisel n comple roots. What is the degree of a quadratic equation? How man comple roots should it have? Stud the graphs of the quadratic functions below. Eplain how their related equations can each have two comple roots. Eplain: Eplain: Eplain: 01, TESCCC 09/06/1 page 3 of 6

4 Sample Problems Solving Quadratic Equations b Graphs and Factoring Complete the tables to determine the zeros of each quadratic function. A) = B) f ( ) = Zeros: Zeros: Use a calculator to sketch the graph of each function. Then tell where the -intercepts occur. C) = D) = E) = Graph: Graph: Graph: Algebra Unit: 05 Lesson: 0 -intercepts: -intercepts: -intercepts: F) Write down the zeros of the quadratic function graphed here. Work backward and use the zeros to write the two factors of the function. Multipl the factors to find an epanded equation for the function. Zeros: Factors: Function: 01, TESCCC 09/06/1 page 4 of 6

5 Solving Quadratic Equations b Graphs and Factoring Part II: Factoring Follow these steps to solve quadratic equations b factoring. 1. Set each equation equal to zero.. Factor. 3. Set each factor equal to zero and solve. 4. Check solutions b graphing the representative function and comparing zeros of the function. Sample Problems A) - = 6 Algebra Unit: 05 Lesson: 0 NOTE: Factoring cannot be used to find irrational or imaginar solutions. B) = 5-9 C) - 4 = , TESCCC 09/06/1 page 5 of 6

6 Solving Quadratic Equations b Graphs and Factoring PRACTICE PROBLEMS Find the zeros of each function using tables and graphs. 1 1) = ) 4 = ) = Table: Graph (sketch): Graph (sketch): Zeros: Zeros: Zeros: Algebra Unit: 05 Lesson: 0 4) Write down the zeros of the quadratic function graphed here. Work backward and use the zeros to write the two factors of the function. Multipl the factors to find an epanded equation for the function. Zeros: Factors: Function: Find the solutions to the quadratic equations b factoring. Check all possible solutions. 5) = 0 6) + 3 = 10 7) = 0 8) 3 + = 01, TESCCC 09/06/1 page 6 of 6