1. Section 6.6 Solving Equations by Factoring

2. Objective 1: Identify a quadratic equation and write it in standard form. 6.6 Lecture Guide: Solving Equations by Factoring

3. Algebraically If a, b, and c are real constants and, then is the _________ form of a quadratic equation in x. Verbally A quadratic equation in x is a ____________ - degree equation in x. Algebraic Example is a quadratic equation with: Quadratic term : Linear term : Constant term : 1 Quadratic Equation

4. 1. Identify which of the following are quadratic equations in one variable. (a) (b)

5. 1. Identify which of the following are quadratic equations in one variable. (c) (d)

6. (a) Standard form: _______________________ 2. Write each quadratic equation in standard form and identify a, b and c.

7. (b) Standard form: _______________________ 2. Write each quadratic equation in standard form and identify a, b and c.

8. (c) Standard form: _______________________ 2. Write each quadratic equation in standard form and identify a, b and c.

9. (d) Standard form: _______________________ 2. Write each quadratic equation in standard form and identify a, b and c.

10. Algebraically If a and b are real numbers: if a = 0, or b = 0, then ab = 0. If ab = 0, then a = 0 or b = 0. Verbally The product of zero and any other factor is ____________. If the product of two factors _________, at least one of the factors must be zero. Algebraic Example If then either x – 3 = 0 or x – 4 = 0. Objective 2: Use factoring to solve selected second- and third-degree equations. Zero-factor Principle

11. Solve each equation. 3.4.

12. Verbally Step 1. Write the equation in ____________ form, with the right side zero. Step 2. ____________ the left side of the equation. Step 3. Set each factor equal to _________. Step 4. Solve the resulting first- degree equations. Algebraic Example Solving Quadratic Equations by Factoring

13. Solve each equation. 5.

14. Solve each equation. 6.

15. Solve each equation. 7.

16. Solve each equation. 8.

17. Solve each equation. 9.

18. Solve each equation. 10.

19. Solve each equation. 11.

20. Solve each equation. 12.

21. Solve each equation. 13.

22. Solve each equation. 14.

23. Solve each equation. 15.

24. Solve each equation. 16.

25. Objective 3: Construct a quadratic equation with given solutions. For a real constant c and a real polynomial, the following statements are equivalent. AlgebraicallyNumericallyGraphically is a factor of is a solution of, that is c is a zero of is an x- intercept of the graph of Equivalent Statements About Linear Factors of a Polynomial

26. Solve 17.

27. Construct a quadratic equation in x with the given solutions. 18. and

28. Construct a quadratic equation in x with the given solutions. 19. and

29. Construct a quadratic equation in x with the given solutions. 20. and

30. Construct a quadratic equation in x with the given solutions. 21. and

31. Construct a quadratic equation in x with the given solutions. 22. and

32. Objective 4: Solve a quadratic inequality by using the x-intercepts of the corresponding graph. Solutions of Equations and Inequalities If is a real polynomial and c is a real number, then: VerballyAlgebraicallyGraphically c is a solution of is an x- intercept of the graph of. c is a solution ofAt, the graph is below the x-axis. c is a solution ofAt, the graph is above the x-axis.

33. 23. Use the graph to help determine the solution of each equation and inequality. (a) (b) (c)

34. 24. Use the graph to help determine the solution of each equation and inequality. (a) (b) (c)

35. (a) Determine the zeros of. (b) Determine the x – intercepts of the graph of. (c) Factor. (d) Give the solutions of. 25. Use the table for to:

36. 26. Solve vs Simplify In part (a), solve the equation. In part (b), perform the indicated operations and simplify the result. (a)

37. 26. Solve vs Simplify In part (a), solve the equation. In part (b), perform the indicated operations and simplify the result. (b)

38. 27. The length of the rectangle shown in the figure is 5 cm less than twice the width. Find the dimensions of this rectangle if the area is 63 cm 2.