Answers to Math 100 Exam 1
1
Grand total for the whole exams: 117 pts.
#1 (total: 7 pts):
(a) (4 pts):
√
Simplify a2 , given that a < 0.
Answer:
√
a2 = |a| because the square root is an even root
= − a because a < 0.
(b) (3 pts):
√
3
Simplify a3 , given that a < 0.
Answer:
√
a3 =a because the cube root is an odd root.
#2 (total: 7 pts):
What is the slope of the line which goes through the points (3, −2) and (−3, −4)? (Simplified
answer please.)
Answer:
m=
−2 − (−4)
−2 + 4
2
1
=
= = .
3 − (−3)
3+3
6
3
#3 (total: 9 pts): What is the equation, in standard form, of the circle of radius
at (−1, 2)?
Answer:
√ 2
(x − (−1))2 + (y − 2)2 =
7 , that is,
(x + 1)2 + (y − 2)2 = 7.
#4 (total: 8 pts): Simplify (rationalize the denominator):
√
3+ 5
√ .
2+ 5
√
7 centered
Answers to Math 100 Exam 1
2
Answer:
√
√
√ √
√
√
√
√
√
3+ 5 2− 5
3·2− 5· 5−3 5+2 5
1− 5
3+ 5
6−5− 5
√ =
√ ·
√ =
=
=
√ 2
4−5
−1
2+ 5
2+ 5 2− 5
22 −
5
√
= −1 + 5.
#5 (total: 4 pts): Simplify
c2q+3
.
(bcq )2
(Assume that q is a natural numbers. Answer should not contain any negative exponent.)
Answer:
c2q+3
c(2q+3)−(2q)
c3
c2q+3
=
=
=
.
b2 c2q
b2
b2
(bcq )2
#7 (total: 6 pts): Simplify
−2 −1 !−1
2
1
+
.
2
3
(Answer should not contain any exponent.)
Answer:
!−1 −2 −1 !−1
2
−1 −1 −1
1
2
2
3
3
8 3
11
2
+
=
+
= 4+
=
=
= .
+
2
3
1
2
2
2 2
2
11
#7 (total: 14 pts): Solve
4x2 + 4x = 3
(simplified answer please).
Answer: Equivalent to
4x2 + 4x − 3 = 0
Answers to Math 100 Exam 1
3
ax2 + bx + c = 0 with a = 4, b = 4 and c = −3.
b2 − 4ac = 42 − 4 · 4 · (−3) = 16 + 48 = 64.
Solutions are

√
√
2
−4 ± 64
−4 ± 8 
−b ± b − 4ac
=
=
=

2a
2·4
8
−4+8
8
=
4
8
−4−8
8
=
−12
8
=
1
2
= − 32
Final answer: x = 1/2 or x = −3/2.
#8 (total: 9 pts): Solve
5x5 − 20x3 = 0
(simplified answer please).
Answer:
Case 1:
5x3 x2 − 4 = 0.
x3 = 0
=⇒
=⇒
x3 = 0 or x2 − 4 = 0.
x = 0.
Case 2:
x2 − 4 = 0
=⇒
(x − 2)(x + 2) = 0
=⇒
x = 2 or x = −2
(Could also be done with the quadratic formula.)
Final answer: x = 0 of x = 2 or x = −2.
#9 (total: 16 pts): Suppose that the graph of
Ax + By + C = 0
is a line which is not vertical.
(a) (5 pts): What is the equation of the line in slope-intercept form y = mx + b?
Answer:
By = −Ax − C
=⇒
y=−
C
A
x− .
B
B
Answers to Math 100 Exam 1
4
(b) (2 pts): What is the slope of the line?
m=−
A
.
B
(c) (2 pts): What is the (y-)intercept of the line?
Answer:
b=−
C
.
B
(d) (7 pts): What is the x-intercept of the line?
Answer:
Ax + B · 0 + C = 0
=⇒
Ax = −C
=⇒
C
x=− .
A
If A = 0 (horizontal line), undefined: no x-intercept unless line is x-axis.
#10 (total: 21 pts):
(a) (13 pts):
Draw
|2x + 4| ≤ 6
on a number line.
Answer:
“≤ 6” so it will be a closed interval [a, b].
|2x + 4| = |2| |x + 2| ≤ 6
=⇒
|x + 2| = |x − −2| ≤ 3,
so the interval is
[−2 − 3, −2 + 3] = [−5, 1] .
The picture looks like the one given as answer for problem 57 in Section 1.2 (the answer is on
page A-35), except that you replace −1 by −5 and 7 by 1 (you may use “full dots” • instead of
“[” and “]”).
Alternate solution method:
“≤ 6” so it will be a closed interval [a, b].
Case 1: 2x + 4 ≤ 6 =⇒ x + 2 ≤ 3 =⇒ x ≤ 1
Case 2: − (2x + 4) ≤ 6 =⇒ 2x + 4 ≥ −6 =⇒ x + 2 ≥ −3
Then proceed as before.
=⇒
x ≥ −5.
Answers to Math 100 Exam 1
5
(b) (8 pts): Draw
|2x + 4| > 6
on a number line. (Hint: Look at part (a).)
Answer: These are the other points on the real line:
(−∞, −5) ∪ (1, ∞) .
The picture looks like the one given as answer for problem 53 in Section 1.2 (the answer is on
page A-35), except that you replace −1 by −5 (you may use “empty dots” ◦ instead of “)” and
“(”).
#11 (total: 16 pts): Solve
x
x
2x
+
= 2
x+1 x−1
x −1
(simplified answer please).
Answer: Multiply both sides by x2 − 1 = (x + 1)(x − 1) (multiplying by 0?):
2x
x
x
2
x
−
1
+
((x + 1)(x − 1)) =
x+1 x−1
x2 − 1
=⇒ LHS = x(x − 1) + x(x + 1) = x2 − x + x2 + x = 2x2 and RHS = 2x
=⇒ 2x2 − 2x = 0
=⇒ x2 − 1x = 0
=⇒ x(x − 1) = 0
=⇒ x = 0 or x = 1.
Check in original equation:
0
0 ? 2·0
+
= 2
Yes.
0+1 0−1
0 −1
1
1 ? 2·0
= 2
+
Undefined.
1+1 1−1
1 −1
Final answer: x = 0.