Section 5.2 Solving Quadratic Equations

1. Section 5.2 Solving Quadratic Equations

2. Classwork: • Factoring Introduction Worksheets • Purple Sweater Method • Factoring Summary Worksheet (5 different methods)

3. Objectives: • Solve quadratic equations by factoring, taking square roots, or by squaring. • Use the distance formula to find the distance between two points.

4. Solving Quadratic Equations You should be familiar with the following four methods for solving quadratic equations: • Factoring • Square root principle • Completing the Square • Quadratic formula

5. First… Factoring

6. FACTORING: If ab = 0, then a = 0 or b = 0. We will be solving quadratics equations of the form: ax2 + bx + c = 0 Examples: x2 + x – 6 = 0 (x - 2)(x + 3) = 0 x + 3 = 0 x – 2 = 0 x = -3 x = 2 x2 – 16 = 0 (x - 4)(x + 4) = 0 x – 4 = 0 x + 4 = 0 x = 4 x = -4

7. Let’s try these examples: 2x² + 9x + 7 = 3 2x² + 9x + 4 = 0 (2x + 1)(x + 4) = 0 2x + 1 = 0 x = - ½ x + 4 = 0 x = - 4

8. Let’s try these examples: 6x² - 3x = 0 3x(2x – 1) = 0 3x = 0 x = 0 2x – 1 = 0 x = ½

9. Factoring • Be sure that when solving these equations, the right side is set equal to zero. • All terms must be collected to one side before factoring.

10. Exit Ticket: In the equation (x – 5)(x + 2) = 8, why is is incorrect to set each factor equal to 8? How would you solve this the correct way?

11. Homework: Solving Equations by Factoring Worksheet

12. Now… The Square Root Property

13. What is It could be 3 or -3. To avoid confusion, 3 is called the principal square root and is indicated by the square root sign. When there is no minus sign in front of the square root, assume it is the principal root. NOTE: has two solutions, 3 and -3. However, and

14. Example: x2 = 9 x2 – 21 = 0 x2 = 21 ≈ x + 3 = ± 4 (x + 3)2 = 16 x = -3 ± 4 x = 1 or x = -7

15. Example: (a) 4x² = 12 x² = 3 x = ±√3 (b) (x – 3)² = 7 x – 3 = ±√7 x = 3 ±√7

16. Example: Solve 4x² + 13 = 253 4x² = 240 x² = 60 SIMPLIFY: This is called the simple radical form.

17. Now… Solving Radical Expressions

18. Solving Radical Equations

19. A radical equation is an equation that contains a radical.

20. The goal in solving radical equations is the same as the goal in solving most equations.

21. We need to isolate the variable.

22. But there is only one way to move the variable out from under the square root sign.

23. We need to square the radical expression.

24. And, because it is an equation, what we do to one side,

25. And, because it is an equation, what we do to one side, we have to do to the other.

26. And now, we need to simplify:

27. Remember, no matter what n is. (Even if n is an expression)

28. So we have: So we have:

29. Another Example:

30. Solve for x: Step 1. Isolate the radical

31. Solve for x: Step 2. Square both sides.

32. Solve for x: Step 3. Set one side equal to 0

33. Solve for x: Step 4. Factor

34. Solve for x: Step 5. Solve the Equation x + 3 = 0 x = -3 x = 1 x – 1 = 0

35. Extraneous Roots • Be careful! Sometimes when you square both sides of an equation you can get extraneous roots, or false roots. • For example, -5 = 5 is false, but (-5)2 = 52 is true. • Therefore, you must check your answer in the original equation when you square both sides.

36. Solve Check your answer!!! There is no solution! Could you tell it had no solution without even solving it? How?

37. Homework • Solving Radical Equations Worksheet

38. In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the legs. c a q b Now… Remember the Pythagorean Theorem????? c2 = a2 + b2

39. Pythagorean Theorem Distance is always positive! Therefore, all negative answers that result from taking the square root will be eliminated!

40. Example: Find the length of the hypotenuse. 4 6 c² = a² + b² c² = 4² + 6² c² = 16 + 36 The length of the hypotenuse is about 7.2

41. Example: Find the length of the missing leg. 5 c² = a² + b² (√74)² = 5² + b² 74 = 25 + b² 49 = b² The length of the leg is 7.

42. The Distance Formula

45. Distance