Quadratics

1. Write x^2 - 10x + 7 in completed-square form.

1. (x - 5)^2 - 18
2. (x - 5)^2 + 7
3. (x - 10)^2 - 93
4. (x + 5)^2 - 18

2. Write 2x^2 + 12x - 1 in completed-square form.

1. 2(x + 3)^2 - 19
2. 2(x + 6)^2 - 73
3. 2(x + 3)^2 - 1
4. (2x + 3)^2 - 10

3. For real x, state the vertex and range of y = -3x^2 + 12x - 5.

1. Vertex (2, 7), range y <= 7
2. Vertex (-2, 7), range y <= 7
3. Vertex (2, -7), range y >= -7
4. Vertex (4, 7), range y <= 7

4. For x >= 1, find the range of f(x) = (x + 2)^2 - 5.

1. f(x) >= 4
2. f(x) >= -5
3. f(x) <= 4
4. -5 <= f(x) <= 4

5. How many real roots does 4x^2 + 4x + 5 = 0 have?

1. No real roots
2. One repeated real root
3. Two distinct real roots
4. Exactly one positive real root

6. Find k if x^2 + kx + 9 = 0 has a repeated real root.

1. k = 6 or k = -6
2. k = 6 only
3. k = -6 only
4. k = 3 or k = -3

7. For which values of m does x^2 + 2x + m = 0 have two distinct real roots?

1. m <= 1
2. m < 1
3. m > 1
4. m >= 1

8. Solve 3x^2 - 11x - 4 = 0.

1. x = -4 or x = 1/3
2. x = 4 or x = -1/3
3. x = 4 or x = 1/3
4. x = -4 or x = -1/3

9. Solve x^2 + 6x - 2 = 0 exactly.

1. x = 3 +/- sqrt(11)
2. x = -3 +/- sqrt(11)
3. x = -6 +/- sqrt(38)
4. x = -3 +/- sqrt(7)

10. Solve 2x^2 + x - 7 = 0 exactly.

1. x = (1 +/- sqrt(57))/4
2. x = (-1 +/- sqrt(57))/4
3. x = (-1 +/- sqrt(29))/4
4. x = (-1 +/- sqrt(57))/2

11. Solve 7x^2 - 5x - 4 = 0, giving both roots to 3 significant figures.

1. x = 1.20 or x = -0.477
2. x = 1.19 or x = -0.479
3. x = 1.19 or x = -0.478
4. x = 0.479 or x = -1.19

12. Solve x^2 - 7x + 10 > 0.

1. 2 < x < 5
2. x < 2 or x > 5
3. x <= 2 or x >= 5
4. 2 <= x <= 5

13. Solve 6 - x - x^2 >= 0.

1. x <= -3 or x >= 2
2. -2 <= x <= 3
3. -3 <= x <= 2
4. x < -3 or x > 2

14. Solve (x - 1)(x + 4) < 6.

1. x < -5 or x > 2
2. -4 < x < 1
3. -5 < x < 2
4. -5 <= x <= 2

15. Solve y = 3x - 2 and y = x^2 - x + 2 simultaneously.

1. (1, 1) and (4, 10)
2. (-2, -8) and (2, 4)
3. (2, 4) only
4. (0, -2) and (4, 10)

16. Solve x + y = 5 and xy = 6 simultaneously.

1. (-2, 7) and (7, -2)
2. (1, 6) and (6, 1)
3. (2, 3) and (3, 2)
4. (2, -3) and (-3, 2)

17. How many real solutions does the simultaneous system y = x + 4 and y = x^2 + 2 have?

1. One
2. None
3. Two
4. Infinitely many

18. Solve x^4 - 10x^2 + 9 = 0 for real x.

1. x = -9, -1, 1 or 9
2. x = 1 or 9
3. x = -3, -1, 1 or 3
4. x = -3 or 3

19. Solve x^(2/3) - 5x^(1/3) + 6 = 0 for real x.

1. x = 4 or x = 9
2. x = -8 or x = -27
3. x = 2 or x = 3
4. x = 8 or x = 27

20. Solve x - 7sqrt(x) + 12 = 0 for real x.

1. x = 3 or x = 4
2. x = -9 or x = -16
3. x = 4 or x = 49
4. x = 9 or x = 16

21. Solve sqrt(x + 6) = x for real x.

1. x = -2 or x = 3
2. x = -3 only
3. x = 2 only
4. x = 3 only

22. For x != 0, solve 1/x^2 - 5/x + 6 = 0.

1. x = 2 or x = 3
2. x = -1/2 or x = -1/3
3. x = 0, 1/2 or 1/3
4. x = 1/2 or x = 1/3

23. The equation x^2 + (p + 1)x + p = 0 has a repeated real root. What is p?

1. p = -1
2. p = 0
3. p = 2
4. p = 1

24. The line y = 2x + k touches the parabola y = x^2 - 4x + 7 at exactly one point. Find k.

1. k = 2
2. k = -7
3. k = 7
4. k = -2