Download Math 125 Practice Test #3 (Chapter 8 and Chapter 9) 1.

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)
− 54a 8b 5
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 + 5 x 3 16 x 2
7.
Multiply.
a)
8.
a +2
)
2
b)
45 x 3 − 18 x 2 + 50 x 2 − 20 x 3
(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.
a)
2x − 3 = 3 − x
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)
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
2
3
2
=
x
1
3
d) x − 6 x =
−5
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 + 3 x − 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 (=
b) g ( x) =
− x2 + 5
x) 2( x + 1) 2
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) =
− x2 + 6x + 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) −3x 12
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) −3a 2b
b)
3
2a 2b 2
1
b) 16x 2
c)
(
3
2x 2
5
a) x = ± i
2
1
c) x = , 4
4
d) 3
1
36
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
± 2, x =
± 2
b) x =
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: h 2 + 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.