# The Quadratic Formula

In Exercises 1-8, find all real solutions

of the given equation. Use a calculator to

approximate the answers, correct to the

nearest hundredth (two decimal places ).

1. x^{2} = 36

2. x^{2} = 81

3. x^{2} = 17

4. x^{2} = 13

5. x^{2} = 0

6. x^{2} = −18

7. x^{2} = −12

8. x^{2} = 3

In Exercises 9-16, find all real solutions

of the given equation. Use a calculator to

approximate your answers to the nearest

hundredth.

9. (x − 1)^{2} = 25

10. (x + 3)^{2} = 9

11. (x + 2)^{2} = 0

12. (x − 3)^{2} = −9

13. (x + 6)^{2} = −81

14. (x + 7)^{2} = 10

15. (x − 8)^{2} = 15

16. (x + 10)^{2} = 37

In Exercises 17-28, perform each of the

following tasks for the given quadratic

function.

i. Set up a coordinate system on a sheet

of graph paper . Label and scale each

axis. Remember to draw all lines with

a ruler .

ii. Place the quadratic function in vertex

form. Plot the vertex on your coordinate

system and label it with its

coordinates. Draw the axis of symmetry

on your coordinate system and

label it with its equation.

iii. Use the quadratic formula to find the

x - intercepts of the parabola . Use a

calculator to approximate each intercept,

correct to the nearest tenth, and

use these approximations to plot the

x-intercepts on your coordinate system.

However, label each x-intercept

with its exact coordinates.

iv. Plot the y-intercept on your coordinate

system and its mirror image across

the axis of symmetry and label each

with their coordinates.

v. Using all of the information on your

coordinate system, draw the graph of

the parabola , then label it with the

vertex form of the function. Use interval

notation to state the domain

and range of the quadratic function.

17. f(x) = x^{2} − 4x − 8

18. f(x) = x^{2} + 6x − 1

19. f(x) = x^{2} + 6x − 3

20. f(x) = x^{2} − 8x + 1

21. f(x) = −x^{2} + 2x + 10

22. f(x) = −x^{2} − 8x − 8

23. f(x) = −x^{2} − 8x − 9

24. f(x) = −x^{2} + 10x − 20

25. f(x) = 2x^{2} − 20x + 40

26. f(x) = 2x^{2} − 16x + 12

27. f(x) = −2x^{2} + 16x + 8

28. f(x) = −2x^{2} − 24x − 52

In Exercises 29-32, perform each of the

following tasks for the given quadratic

equation.

i. Set up a coordinate system on a sheet

of graph paper. Label and scale each

axis. Remember to draw all lines with

a ruler .

ii. Show that the discriminant is negative .

iii. Use the technique of completing the

square to put the quadratic function

in vertex form. Plot the vertex on

your coordinate system and label it

with its coordinates. Draw the axis of

symmetry on your coordinate system

and label it with its equation.

iv. Plot the y-intercept and its mirror

image across the axis of symmetry

on your coordinate system and label

each with their coordinates.

v. Because the discriminant is negative

(did you remember to show that?),

there are no x-intercepts. Use the

given equation to calculate one additional

point, then plot the point and

its mirror image across the axis of

symmetry and label each with their

coordinates.

vi. Using all of the information on your

coordinate system, draw the graph of

the parabola, then label it with the

vertex form of function. Use interval

notation to describe the domain and

range of the quadratic function.

29. f(x) = x^{2} + 4x + 8

30. f(x) = x^{2} − 4x + 9

31. f(x) = −x^{2} + 6x − 11

32. f(x) = −x^{2} − 8x − 20

In Exercises 33-36, perform each of the

following tasks for the given quadratic

function.

i. Set up a coordinate system on a sheet

of graph paper. Label and scale each

axis. Remember to draw all lines with

a ruler.

ii. Use the discriminant to help determine

the value of k so that the graph

of the given quadratic function has

exactly one x-intercept.

iii. Substitute this value of k back into

the given quadratic function, then use

the technique of completing the square

to put the quadratic function in vertex

form. Plot the vertex on your coordinate

system and label it with its

coordinates. Draw the axis of symmetry

on your coordinate system and

label it with its equation.

iv. Plot the y-intercept and its mirror

image across the axis of symmetry

and label each with their coordinates.

v. Use the equation to calculate an additional

point on either side of the axis

of symmetry, then plot this point and

its mirror image across the axis of

symmetry and label each with their

coordinates.

vi. Using all of the information on your

coordinate system, draw the graph

of the parabola, then label it with

the vertex form of the function. Use

interval notation to describe the domain

and range of the quadratic function.

33. f(x) = x^{2} − 4x + 4k

34. f(x) = x^{2} + 6x + 3k

35. f(x) = kx^{2} − 16x − 32

36. f(x) = kx^{2} − 24x + 48

37. Find all values of k so that the graph

of the quadratic function f(x) = kx^{2} −

3x + 5 has exactly two x-intercepts.

38. Find all values of k so that the graph

of the quadratic function f(x) = 2x^{2} +

7x − 4k has exactly two x-intercepts.

39. Find all values of k so that the graph

of the quadratic function f(x) = 2x^{2} −

x + 5k has no x-intercepts.

40. Find all values of k so that the graph

of the quadratic function f(x) = kx^{2} −

2x − 4 has no x-intercepts.

In Exercises 41-50, find all real solutions,

if any, of the equation f(x) = b.

41. f(x) = 63x^{2} + 74x − 1; b = 8

42. f(x) = 64x^{2} + 128x + 64; b = 0

43. f(x) = x^{2} − x − 5; b = 2

44. f(x) = 5x^{2} − 5x; b = 3

45. f(x) = 4x^{2} + 4x − 1; b = −2

46. f(x) = 2x^{2} − 9x − 3; b = −1

47. f(x) = 2x^{2} + 4x + 6; b = 0

48. f(x) = 24x^{2} − 54x + 27; b = 0

49. f(x) = −3x^{2} + 2x − 13; b = −5

50. f(x) = x^{2} − 5x − 7; b = 0

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