The document discusses the quadratic formula and how to use the discriminant to determine the number of solutions a quadratic equation has. It provides examples of calculating the discriminant for different quadratic equations and relating the discriminant to whether there are 0, 1, or 2 solutions graphically.

1. Quadratic Formula -b + b 2 - 4ac 2a a x 2 + b x + c = 0 Rewrite quadratic equation in standard form a substitute a , b , and c into formula. x =

2. Solve: 3x 2 - 5 = 2x

3. Step 1: Rewrite in standard form. Solve: 3x 2 - 5 = 2x

4. Solve: 3x 2 - 5 = 2x Step 1: Rewrite in standard form. 3x 2 - 2x - 5 = 0 a b c

5. Solve: 3x 2 - 5 = 2x Step 1: Rewrite in standard form. Step 2: Plug into formula and simplify. 3x 2 - 2x - 5 = 0 a b c

6. Solve: 3x 2 - 5 = 2x Step 1: Rewrite in standard form. Step 2: Plug into formula and simplify. 3x 2 - 2x - 5 = 0 a b c x = -(-2) + (-2) 2 - 4(3)(-5) 2(3) x = -b + b 2 - 4ac 2a

7. Solve: 3x 2 - 5 = 2x Step 1: Rewrite in standard form. Step 2: Plug into formula and simplify. 3x 2 - 2x - 5 = 0 a b c x = -(-2) + (-2) 2 - 4(3)(-5) 2(3) x = 2 + 64 6 x = -b + b 2 - 4ac 2a

8. Solve: 3x 2 - 5 = 2x Step 1: Rewrite in standard form. Step 2: Plug into formula and simplify. 3x 2 - 2x - 5 = 0 a b c x = -(-2) + (-2) 2 - 4(3)(-5) 2(3) x = 2 + 64 6 x = 2 + 8 6 x = -b + b 2 - 4ac 2a

9. Solve: 3x 2 - 5 = 2x Step 1: Rewrite in standard form. Step 2: Plug into formula and simplify. = 5/3, -1 3x 2 - 2x - 5 = 0 a b c x = -(-2) + (-2) 2 - 4(3)(-5) 2(3) x = 2 + 64 6 x = 2 + 8 6 x = -b + b 2 - 4ac 2a

10. Solve: 3x 2 = 5x - 2

11. Solve: 3x 2 = 5x - 2 3x 2 - 5x + 2 = 0

12. Solve: 3x 2 = 5x - 2 3x 2 - 5x + 2 = 0 x = -(-5) + (-5) 2 - 4(3)(2) 2(3)

14. Solve: 3x 2 = 5x - 2 3x 2 - 5x + 2 = 0 = 1, 2/3 x = -(-5) + (-5) 2 - 4(3)(2) 2(3) x = 5 + 1 6

15. Discriminant - used to determine whether the quadratic equation has 0, 1, or 2 answers. d = b 2 - 4ac

16. Discriminant - used to determine whether the quadratic equation has 0, 1, or 2 answers. d = b 2 - 4ac Where do you recognized this formula from?

17. Discriminant - used to determine whether the quadratic equation has 0, 1, or 2 answers. d = b 2 - 4ac Where do you recognized this formula from? It’s part of the quadratic formula.

18. Discriminant - used to determine whether the quadratic equation has 0, 1, or 2 real number solutions. d = b 2 - 4ac Where do you recognized this formula from? It’s part of the quadratic formula. If d = 0, then there is exactly one solution x = -b + b 2 - 4ac 2a

19. Discriminant - used to determine whether the quadratic equation has 0, 1, or 2 real number solutions. d = b 2 - 4ac Where do you recognized this formula from? It’s part of the quadratic formula. If d = 0, then there is exactly one solution If d > 0, then there are two solutions. x = -b + b 2 - 4ac 2a

20. Discriminant - used to determine whether the quadratic equation has 0, 1, or 2 real number solutions. d = b 2 - 4ac Where do you recognized this formula from? It’s part of the quadratic formula. If d = 0, then there is exactly one solution If d > 0, then there are two solutions. If d < 0, then there are no real number solutions. x = -b + b 2 - 4ac 2a

21. How many solutions does x 2 - 2x + 1 = 0 have? d = b 2 - 4ac

22. How many solutions does x 2 - 2x + 1 = 0 have? d = b 2 - 4ac d = (-2) 2 - 4(1)(1)

23. How many solutions does x 2 - 2x + 1 = 0 have? d = b 2 - 4ac d = (-2) 2 - 4(1)(1) d = 4 - 4

24. How many solutions does x 2 - 2x + 1 = 0 have? d = b 2 - 4ac d = (-2) 2 - 4(1)(1) d = 4 - 4 d = 0

25. How many solutions does x 2 - 2x + 1 = 0 have? d = b 2 - 4ac d = (-2) 2 - 4(1)(1) d = 4 - 4 d = 0 One solution

26. How many solutions does x 2 - 2x + 1 = 0 have? d = b 2 - 4ac d = (-2) 2 - 4(1)(1) d = 4 - 4 d = 0 One solution Let’s look at the equation graphically.

27. How many solutions does x 2 - 2x + 1 = 0 have? d = b 2 - 4ac d = (-2) 2 - 4(1)(1) d = 4 - 4 d = 0 One solution Let’s look at the equation graphically.

28. How many solutions does x 2 - 2x + 1 = 0 have? d = b 2 - 4ac d = (-2) 2 - 4(1)(1) d = 4 - 4 d = 0 One solution Let’s look at the equation graphically. Intersects x-axis only once.

29. How many solutions does 2x 2 - 2x + 1 = 0 have?

30. How many solutions does 2x 2 - 2x + 1 = 0 have? d = (-2) 2 - 4(2)(1)

31. How many solutions does 2x 2 - 2x + 1 = 0 have? d = (-2) 2 - 4(2)(1) d = 4 - 8 d = -4

32. How many solutions does 2x 2 - 2x + 1 = 0 have? d = (-2) 2 - 4(2)(1) d = 4 - 8 d = -4 No Real Number Solutions

33. How many solutions does 2x 2 - 2x + 1 = 0 have? d = (-2) 2 - 4(2)(1) d = 4 - 8 d = -4 No Real Number Solutions Graphically:

34. How many solutions does 2x 2 - 2x + 1 = 0 have? d = (-2) 2 - 4(2)(1) d = 4 - 8 d = -4 No Real Number Solutions Graphically: Does not cross x-axis

35. How many solutions does 3x 2 - 2x - 1 = 0 have?

36. d = (-2) 2 - 4(3)(-1) How many solutions does 3x 2 - 2x - 1 = 0 have?

37. d = (-2) 2 - 4(3)(-1) How many solutions does 3x 2 - 2x - 1 = 0 have? d = 4 + 12 d = 16

38. d = (-2) 2 - 4(3)(-1) How many solutions does 3x 2 - 2x - 1 = 0 have? d = 4 + 12 d = 16 Two solutions

39. d = (-2) 2 - 4(3)(-1) How many solutions does 3x 2 - 2x - 1 = 0 have? d = 4 + 12 d = 16 Two solutions Graphically: Crosses x-axis twice