Math 152
Chapter 8 Handout
Helene Payne
Name:
1. Solve the equation by the square root property. If possible, simplify radicals or rationalize denominators. Express imaginary solutions in the form
a + bi.
(a) x2 = 25
(b) 4x2 = 400
(c) 16x2 + 49 = 0
(d) (x + 2)2 = 49
2
1
121
(e) x −
=
2
4
2. Complete the square for the binomial. Then factor the resulting perfect
square trinomial.
(a) x2 + 10x
(b) x2 − 18x
Math 152
Chapter 8 Handout
Helene Payne
3. Solve the quadratic equation by completing the square.
(a) x2 − 8x + 15 = 0
(b) x2 + 14x + 34 = 0
4. A ladder that is 5 feet long is 3 feet from the base of the wall. How far
up the wall does the ladder reach? Solve this problem using only one
variable. Draw a picture to assist you.
5. The function s(t) = 16t2 models the distance, s(t), in feet, that an object
falls in t seconds. Find the number of seconds a sky diver is in free
fall after jumping from a plane and falling 704 feet before opening a
parachute. Express answers in simplified radical form.
Page 2
Math 152
Chapter 8 Handout
Helene Payne
6. Solve each equation by using the quadratic formula. Simplify solutions if
possible.
If ax2 + bx + c = 0, then the equation can be solved by using the quadratic
formula: √
−b ± b2 − 4ac
x=
2a
2
(a) x − 14x + 40 = 0
(b) 2x2 + x − 21 = 0
(c) 3x2 = 7
7. Compute the discriminant (b2 − 4ac), then determine whether the equation has (i) two rational solutions, (ii) two irrational solutions, (iii) one
real solution, or (iv) two imaginary solutions.
(a) x2 − 4x + 4 = 0
(b) x2 + 6x + 10 = 0
(c) 2x2 − 7x + 1 = 0
Page 3
Math 152
Chapter 8 Handout
Helene Payne
8. Solve the equation by the method of your choice. Simplify solutions if
possible.
(a) (x − 6)2 = 17
(b) x2 + 4x = 0
(c) 4x2 + 6x + 1 = 0
(d) 3x2 − 6x + 4 = 0
9. Write a quadratic equation in standard form with the given solution set.
(a) {−4, 10}
√ √
(b) {−2 3, 2 3}
(c) {−7i, 7i}
Page 4
Math 152
Chapter 8 Handout
Helene Payne
10. A rectangular sign must have an area of 45 yards. Its length must be 4
yards more than its width. Find the dimensions of the sign. Round to
the nearest tenth of a yard, and solve using only one variable.
11. Find the coordinates of the vertex for the parabola defined by the given
quadratic function.
(a) f (x) = x2 + 4
(b) f (x) = (x + 5)2 − 7
(c) f (x) = −6x2 + 12x + 3
Page 5
Math 152
Chapter 8 Handout
Helene Payne
12. Below are the graphs of four quadratic functions. Match the functions
with the correct graph.
(a) f (x) = (x + 1)2
(c) h(x) = (x − 1)2
(b) g(x) = x2 + 1
(d) w(x) = x2 − 1
function:
function:
function:
function:
Page 6
Math 152
Chapter 8 Handout
Helene Payne
13. Determine whether the given quadratic function has a (i) minimum or
maximum value, (ii) find the coordinates of the minimum or maximum
point, and (iii) identify the function’s domain and range.
(a) f (x) = x2 + 2x − 2
minimum / maximum (circle one)
coordinates of minimum / maximum point: ( , )
domain (on interval notation):
range (on interval notation):
(b) f (x) = −x2 + 4x
minimum / maximum (circle one)
coordinates of minimum / maximum point: ( , )
domain (on interval notation):
range (on interval notation):
(c) f (x) = 2x2 − 8x + 16
minimum / maximum (circle one)
coordinates of minimum / maximum point: ( , )
domain (on interval notation):
range (on interval notation):
Page 7
Math 152
Chapter 8 Handout
Helene Payne
14. Only one variable should be used to solve these word problems. I recommend solving it using the five steps method.
(a) Among all pairs of numbers whose sum is 30, find a pair whose
product is as large as possible.
(b) A person standing close to the edge of an 88-foot building throws a
baseball vertically upward. The quadratic function s(t) = −16t2 +
64t + 88 models the ball’s height above the ground, s(t), in feet, t
seconds after it was thrown. How many seconds does it take until
the ball finally hits the ground? Round to the nearest tenth of a
second if necessary.
Page 8
Math 152
Chapter 8 Handout
Helene Payne
15. Solve the equations below by making an appropriate substitution.
(a) x4 − 40x2 + 144 = 0
1
(b) x − 16x 2 − 512 = 0
(c) (x − 4)2 + 3(x − 4) − 18 = 0
(d) 2x−2 − x−1 − 1 = 0
√
(e) x − 3 x = −2
Page 9
Math 152
Chapter 8 Handout
Helene Payne
16. Solve the quadratic inequality. Graph the solution set on a numberline,
then give the solution in interval notation.
(a) x2 + 2x − 15 > 0
Graph the solution on a number line:
x
Solution on interval notation:
(b) (5x − 2)(x + 7) < 0
Graph the solution on a number line:
x
Solution on interval notation:
(c) 6x2 − 5x + 1 ≤ 0
Graph the solution on a number line:
x
Solution on interval notation:
Page 10
Math 152
Chapter 8 Handout
Helene Payne
17. Solve the rational inequality. Graph the solution set on a numberline,
then give the solution in interval notation.
x+2
(a)
<0
x+4
Graph the solution on a number line:
x
Solution on interval notation:
x
(b)
>2
x−4
Graph the solution on a number line:
x
Solution on interval notation:
x−3
≤3
(c)
x+3
Graph the solution on a number line:
x
Solution on interval notation:
Page 11
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