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Solving Quadratics. Sometimes solving quadratics is easy. Sometimes you recognize a form. Sometimes you can factor. But no matter what, You can ALWAYS use the Quadratic Formula. Example. What does the QF say?. What does the QF say?. What does the QF say?. QF says. QF says.

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2. Sometimes solving quadratics is easy

3. Sometimes you recognize a form

4. Sometimes you can factor

5. But no matter what,You can ALWAYS use the Quadratic Formula

6. Example

7. What does the QF say?

8. What does the QF say?

9. What does the QF say?

10. QF says

11. QF says Location of x-intercept, “roots” or “zeros” of the parabola Line of symmetry, Location of the vertex, Location of the max or min

12. Synonyms

13. Synonyms Line of symmetry Location of vertex Location of extremum (max or min) x-intercept root zero x-intercept root zero Vertex Extremum

14. Why does the QF work? ax b ax2 bx D x + x =

15. Stretch everything by a b ax (ax)2 abx aD + = ax ax

16. Split b in half ax b/2 b/2 (ax)2 abx /2 abx /2 aD + = ax ax

17. Rearrange ax b/2 (ax)2 abx /2 aD = ax abx /2 b/2

18. Complete the square ax b/2 (ax)2 abx /2 aD = b2/4 + ax abx /2 b2/4 b/2

19. Reorganize ax+b/2 (ax+b/2)2 aD = b2/4 + ax +b/2

20. Equationify ax+b/2 (ax+b/2)2 aD = b2/4 + ax +b/2

21. Rearrange ax+b/2 (ax+b/2)2 aD = b2/4 + ax +b/2

22. Square root ax+b/2 (ax+b/2)2 aD = b2/4 + ax +b/2

23. Rearrange ax+b/2 (ax+b/2)2 aD = b2/4 + ax +b/2

24. -b/2 from both sides ax+b/2 (ax+b/2)2 aD = b2/4 + ax +b/2

25. /a on both sides ax+b/2 (ax+b/2)2 aD = b2/4 + ax +b/2

26. But what happened to c?

27. But what happened to c? ax b ax2 bx D x + x =

28. But what happened to c? ax b ax2 bx D x + x - =0

29. But what happened to c? +c=-d ax b ax2 bx D x + x - =0

30. The Quadratic Formula ax2 bx c + + =0

31. Solve: x2+9x+8=0. Select the most correct answer below! • x=1, x=8 • x= -1, x= -8 • x= -1, x = 8 • x=1, x= -8 • No real solutions

32. Find the zeros of f(x)=x2+4x+2 • -2 ± 2sqrt(2) • -2 ± sqrt(2) • -2 ± sqrt(8) • 2 ± 2sqrt(2) • No real solutions

33. Counting Roots (x-2)(x-4) has two real roots: x=2 and x=4.

34. Counting Roots (x-3)(x-3) has two real roots: x=3 and x=3. Both roots are in the same place, But it is useful to think of them as two roots.

35. Counting Roots (x-(3-i))(x-(3+i)) has two complex roots: x=3-i and x=3+i.

36. Counting roots • A quadratic always has exactly two roots • Sometimes the roots are the same • Sometimes the roots are complex • A quadratic always has an even number of complex roots. • Possible roots are: two real, or two complex. You can never have 1 real and 1 complex

37. Why? A quadratic turns and continues infinitely. Because of this, if the quadratic crosses the x axis once, it HAS to cross a second time. Always zero or two real roots.

38. Consider the quadratic function f(x)=x2+2x+5.Which of the following statements is true? • f(x) has 1 real zero and 1 complex zero. • f(x) has no real zeros. • f(x) has 2 real zeros. • f(x) has 3 real zeros. • None of the above are true.

39. Consider the quadratic function f(x)=x2+2x+5.Which of the following statements is true? B) No real zeros

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