Download Chapters 8-9

Math 125 Practice Test #3 (Chapter 8 and Chapter 9)
1.
2.
Evaluate the following radicals with real numbers, if possible.
a) 3 64
b)  16
c) 5  32
d)
b)
 54a8b5
3
Evaluate the given expressions.
5
a)  16 4
4.
81
Simplify the following radicals, if possible. Assume all variables represent positive
real numbers.
a)  75k 7 q10
3.
4
b)
 216 

2
3
Use properties of exponents and simplify the expression. Express your answers with
positive exponents.
1
1
4
a)  3 x x
2
3
(2 x 5 ) 4
b)
x
3
10
4
 m 2 / 3 
c)  3 / 4  m 3 / 8 a 1 / 4
a



2



5.
Find the distance and midpoint between the two points  4 3 , 2 5 and 2 3 ,10 5
6.
Add or subtract.
a) 4 3 54 x 5  5x 3 16 x 2
7.
Multiply.
a)
8.
a 2

2
b)
45x3  18x 2  50 x 2  20 x3
2
3 6

34 6

Rationalize the denominators.
a)
9.
3
b)

3
5 18
3 12
b)
3
3x
4y
4
Solve each equation.
2x  3  3  x
a)
c)
3
11  5
b)
x 
d)
x  4
x  2
x5  5
10.
Perform each operation and express the result in the standard form of a complex number.
 5i
a) 8  4i 7  2i 
b)
2  4i
11.
Multiply.
12.
Solve using the square root property.
a) a 2  40
2 6
b) ( x  3)2  49
13.
14.
15.
16.
Solve by completing the square.
a) x 2  6 x  11  0
b) 2 x 2  5 x  1  0
Solve by using the quadratic formula.
a) 2 x 2  6 x  1
b) 9a 2  4  2a
Solve by factoring.
 2 x  3
x
a) x 2  3 x  18
b)
Solve the given equations.
a) 4 x 2  25  0
b) x 4  6 x 2  8  0
c) 2 x  5 x  2  0
d) x 3  6 x 3  5
2
2
1
17.
Find the discriminant and determine the number and type of solution of the quadratic equation.
a) 9 x 2  42 x  49  0
b) 8 x 2  18 x  5
18.
Solve the inequality, graph and write your answer in interval notation.
a) x 2  3x  10  0
b) x7 x  40 < 12
c)
3x  1
0
x2
d)
x2
 2
x
19.
At a point 16 feet from the base of a tree, the distance to the top of the tree is 4 feet more
than the height h of the tree. Find the height of the tree.
20.
Graph the given quadratic functions. Find the vertex, the x and y-intercepts, and the axis
of symmetry and label them. Which way does the parabola open?
a) f ( x)  2( x  1)2
b) g ( x)   x 2  5
c)
h( x )  2 x 2  8 x  9
21.
Graph the given quadratic functions by completing the square first and then finding other
key points.
a) f ( x)  x 2  2 x  8
b) h( x)   x 2  6 x  9
22.
If an object is thrown upward with an initial velocity of 384 ft/second, then its height y
after t seconds is given by the equation y  384t  32t 2 .
a) Find the maximum height attained by the object.
b) Find the number of seconds it takes the object to hit the ground.
__________________________________________________________________________________
Answer key is on the next page.
Answer Key:
1.
a) 4
2.
a) 5k 3 q 5
3.
a) 32
4.
a) 3x12
5.
Distance = 2 107 ; Midpoint =  3, 6 5
6.
a) 22 x
7.
a) 9a  12 a  4
8.
5 6
a)
6
9.
a) x  2
10.
a) 48  44i
11.
2 3
12.
a) a   2i 10
b) x   4, 10
13.
a) x  3  i 2
b) x 
5  33
4
14.
a) x 
3 7
2
b) a 
1  i 35
9
15.
a) x   6,3
3k
b)
1
36
1
b) 16x 2
c)

3
2x 2
5
a) x   i
2
1
c) x  , 4
4
d) 3
b) 3a 2b 3 2a 2b2
11
16.
c) 2
b) Not a real number
a5/ 2
m 23 / 12

b) x 5 x  2 x 2
b) 30  27 2
3
b)
6 xy 2
2 y2
c)
11  5
2
b) x  4
1
b) 1  i
2
b) x  1,
9
4
b) x   2, x   2
d) x  1, x 125
d)
x 2
17.
a) D  0 so 1 real solution
18.
a) (, 5]  [2, )
c)
19.
 , 2   
1 
,
3 
b) D  484  0 so 2 real solutions
2

b)  6, 
7

 2 
d)   , 0 
 3 
By the Pythagorean theorem: h2  162  (h  4)2  The height of the tree is 30 feet.
20 a). Vertex: (1, 0) ; x-intercept: (1, 0) ; y-intercept: (0, 2); Axis of symmetry: x  1 ; opens up
20 b). Vertex: (0,5)
x-intercept: (  5, 0)
y-intercept: (0, 5)
Axis of symmetry: x  0
opens down
20 c). Vertex: (2,1)
x-intercept: None
y-intercept: (0, 9)
Axis of symmetry: x  2
opens up
21 a).
f ( x)  ( x  1)2  9 ;
Vertex: (1, 9)
f (0)  8 i.e. (0, 8) is the y-intercept
f (4)  0 and f (2)  0 i.e.
(4, 0) and ( 2, 0) are the x-intercepts
f (2)  8 ;
Axis x  1
21 b).
f ( x)  ( x  3)2  18 ; Vertex: (3,18)
f (0)  9 i.e. (0,9) is the y-intercept
(3  3 2, 0) are the x-intercepts, but they are not useful for graphing
f (6)  9 ;
Axis x  3
22 a). The maximum height attained by the object is 1152 feet.
22 b). It takes the object 12 seconds to hit the ground.