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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. x2 = 36
2. x2 = 81
3. x2 = 17
4. x2 = 13
5. x2 = 0
6. x2 = −18
7. x2 = −12
8. x2 = 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) = x2 − 4x − 8
18. f(x) = x2 + 6x − 1
19. f(x) = x2 + 6x − 3
20. f(x) = x2 − 8x + 1
21. f(x) = −x2 + 2x + 10
22. f(x) = −x2 − 8x − 8
23. f(x) = −x2 − 8x − 9
24. f(x) = −x2 + 10x − 20
25. f(x) = 2x2 − 20x + 40
26. f(x) = 2x2 − 16x + 12
27. f(x) = −2x2 + 16x + 8
28. f(x) = −2x2 − 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) = x2 + 4x + 8
30. f(x) = x2 − 4x + 9
31. f(x) = −x2 + 6x − 11
32. f(x) = −x2 − 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) = x2 − 4x + 4k
34. f(x) = x2 + 6x + 3k
35. f(x) = kx2 − 16x − 32
36. f(x) = kx2 − 24x + 48

37. Find all values of k so that the graph
of the quadratic function f(x) = kx2 −
3x + 5 has exactly two x-intercepts.

38. Find all values of k so that the graph
of the quadratic function f(x) = 2x2 +
7x − 4k has exactly two x-intercepts.

39. Find all values of k so that the graph
of the quadratic function f(x) = 2x2 −
x + 5k has no x-intercepts.

40. Find all values of k so that the graph
of the quadratic function f(x) = kx2 −
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) = 63x2 + 74x − 1; b = 8
42. f(x) = 64x2 + 128x + 64; b = 0
43. f(x) = x2 − x − 5; b = 2
44. f(x) = 5x2 − 5x; b = 3
45. f(x) = 4x2 + 4x − 1; b = −2
46. f(x) = 2x2 − 9x − 3; b = −1
47. f(x) = 2x2 + 4x + 6; b = 0
48. f(x) = 24x2 − 54x + 27; b = 0
49. f(x) = −3x2 + 2x − 13; b = −5
50. f(x) = x2 − 5x − 7; b = 0

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