1. Quadratic Equations & Functions

2. Quadratic Equations have x 2 (or some variable, squared) in them and are equations. x 2 + 5x + 6 = 0 n 2 – 7n = 18 2x 2 = 11x + 40 p 2 = 16 (x + 7)(x + 2) = 0

3. The key to solving quadratic equations is the 0 property of multiplication  If the product of two quantities is 0, then one of those quantities must be 0.

4. So, if (x + 3)(x – 2) = 0, then either x + 3 = 0 or x – 2 = 0

5. So, if (x + 3)(x – 2) = 0, then either x + 3 = 0 or x – 2 = 0 This means either x = -3 orx = 2

6. If a quadratic equation is factored, and says (__)(__) = 0:  The answers are the opposite of the factors.

7. Solve (x + 7)(x – 1) = 0 (x – 4)(x – 9) = 0 (x + 8)(x + 13) = 0

8. Solve (x + 7)(x – 1) = 0 x = -7 or 1 (x – 4)(x – 9) = 0 x = 4 or 9 (x + 8)(x + 13) = 0 x = -8 or -13

9. What about these? x(x + 9) = 0 (3x + 7)(x – 2) = 0 (5x – 1)(2x + 1) = 0

10. x(x + 9) = 0 Either x = 0 or x + 9 = 0 So x = 0 or x = -9

11. (3x + 7)(x – 2) = 0 Either 3x + 7 = 0 or x – 2 = 0 So x = -7 / 3 or x = 2

12. (5x – 1)(2x + 1) = 0 Either 5x – 1 = 0 or 2x + 1 = 0 x = 1 / 5 or x = -½

13. When there are coefficients, you can find the answers quickly by making a fraction, reading the factor backwards. (9x – 13)(2x + 7) = 0 x = 13 / 9 or x = -7 / 2

14. If the equation isn’t factored. 1.Write it so it says ___ = 0. 2.Factor. 3.Do the opposite of the factors.

15. n 2 – 7n = 18

16. n 2 – 7n = 18 n 2 – 7n – 18 = 0

17. n 2 – 7n = 18 n 2 – 7n – 18 = 0 (n – 9)(n + 2) = 0

18. n 2 – 7n = 18 n 2 – 7n – 18 = 0 (n – 9)(n + 2) = 0 n = 9 or -2

19. x 2 + 5x + 6 = 0

20. x 2 + 5x + 6 = 0 Already says __ = 0 (x + 2)(x + 3) = 0

21. x 2 + 5x + 6 = 0 Already says __ = 0 (x + 2)(x + 3) = 0 x = -2 or x = -3

22. 2x 2 = 11x + 40

23. 2x 2 = 11x + 40 2x 2 – 11x – 40 = 0

24. 2x 2 = 11x + 40 2x 2 – 11x – 40 = 0 (2x + 5)(x – 8) = 0

25. 2x 2 = 11x + 40 2x 2 – 11x – 40 = 0 (2x + 5)(x – 8) = 0 x = -2 / 5 or x = 8

26. p 2 = 16

27. p 2 = 16 p 2 – 16 = 0

28. p 2 = 16 p 2 – 16 = 0 (p + 4)(p – 4) = 0 p = 4 or -4

29. If a quadratic equation doesn’t factor, there are many other ways it could possibly be solved. You’ll learn many of these in Geometry and in Advanced Algebra.

30. Quadratic Function  Equation always has the form f(x) = ax 2 + bx + c  The simplest quadratic function is f(x) = x 2

31. Graph f(x) = x 2 Like the absolute value function, this gives the same answers for positive and negative numbers.

32. However it is curved at the bottom rather than making a straight V-shape.

33. This U-shaped graph is called a parabola.  Many things in the real world form parabola (arch) shapes.

36. Quadratic functions are used particularly in problems involving  Movement and the force of gravity  Area

37. What we usually care about with quadratic functions are the roots.  These are the places where the function = 0  They can also be called “zeros” or “x-intercepts”.

38. The roots of this quadratic function are -3 and 1. This is the same as the solutions to -x – 2x + 3 = 0

39. Find the roots.

41. Parabolas may have 2, 1, or 0 roots.

42. Find the roots of y = (x + 3)(x – 9) f(x) = (x – 5)(x + 3) g(x) = (x + 2) 2 Just set the functions = 0

43. Find the roots of y = (x + 3)(x – 9) -3 and 9 f(x) = (x – 5)(x + 3) 5 and -3 g(x) = (x + 2) 2 -2

44. Find the roots: f(x) = x 2 + 11x + 28 y = x 2 – 11x + 18 g(x) = x 2 + 1x – 12

45. Find the roots: f(x) = x 2 + 11x + 28 (x + 7)(x + 4) -7 and -4 y = x 2 – 11x + 18 (x – 9)(x – 2) 9 and 2

46. g(x) = x 2 + 1x – 12 (x + 4)(x – 3) -4 and 3

47. Find the roots y = x(x – 8)

48. Find the roots y = x(x – 8) 0 and 8