Workshop #1
Professor D. Olles
1. Simplify the following expressions:
a. 163/4
√
√
b. 200 − 8
c.
1
3
d.
3
π/2
+3−
5
6
2. Factor completely:
a. x2 − 7x + 10
b. 2x3 + 6x2 − 36x
c. 3x2 − 11x − 4
d. 4x2 − 25
e. x3 + 3x2 − 2x − 6
2
3. Simplify x2x−4
4. Solve 2x−3
x+1
x+1
x+2
≤ 1 and express the solution in interval notation.
5. True or False:
√
6. Rationalize
−
3−xy
x
4+h−2
h
=3−y
and simplify the result.
7. Given f (x) = 3x2 − 3x + 5, evaluate the following expressions. Expand
and simplify when possible.
a. f (x + h)
b. f (x) − f (x + h)
c.
f (x)−f (x+h)
h
8. A box with no top is to be created from a 16 x 25 piece of cardboard by
cutting out equal squares from all four corners and folding up the sides.
Write an expression for the volume of the box, V (x).
9. Derive the quadratic formula.
1
Solutions
1. Simplify the following expressions:
a. 163/4
b.
c.
d.
√
1
3
200 −
+3−
161/4
3
= 23 = 8
√
8
√
100 · 2 −
5
6
√
√
√
√
4 · 2 = 10 2 − 2 2 = 8 2
2 18 5
15
5
+
− =
=
6
6
6
6
2
3
π/2
3·
2
6
=
π
π
2. Factor completely:
a. x2 − 7x + 10
(x − 5) (x − 2)
b. 2x3 + 6x2 − 36x
2x x2 + 3x − 18 = 2x (x + 6) (x − 3)
c. 3x2 − 11x − 4
a · c = (3)(−4) = −12
−12 = (−12)(1)
−11 = −12 + 1
3x2 − 11x − 4 = 3x2 − 12x + x − 4
= 3x2 − 12x + (x − 4)
= 3x (x − 4) + (x − 4)
= (x − 4) (3x + 1)
2
d. 4x − 25
(2x − 5) (2x + 5)
e. x3 + 3x2 − 2x − 6
x3 + 3x2 + (−2x − 6)
= x2 (x + 3) + (−2) (x + 3)
= (x + 3) x2 − 2
√ √ = (x + 3) x − 2 x + 2
2
3. Simplify
x2
x2 −4
−
x+1
x+2
x = −1
=
=
x2
x+1
−
(x − 2)(x + 2) x + 2
x2
(x + 1)(x − 2)
−
(x − 2)(x + 2) (x + 2)(x − 2)
=
x2 − (x + 1)(x − 2)
(x + 2)(x − 2)
=
x2 − (x2 − x − 2)
(x + 2)(x − 2)
x2 − x2 + x + 2
(x − 2)(x + 2)
=
=
x+2
(x − 2)(x + 2)
=
4. Solve 2x−3
x+1
1
x−2
≤ 1 and express the solution in interval notation.
2x − 3
=1
x+1
2x − 3 = x + 1
x = 4 critical value
Choose test values x = 0 and x = 5.
x = −1 x = 0
x=4 x=5
-1
0
1
2
3
4
(−∞, −1)
5. True or False:
FALSE
3−xy
x
=3−y
3
[
5
(−1, 4]
6
7
8
h
and simplify the result.
6. Rationalize √4+h−2
√
h
4+h+2
√
√
4+h−2
4+h+2
√
h 4+h+2
= √
√
√
2
4+h +2 4+h−2 4+h−4
√
h 4+h+2
=
4+h−4
√
h 4+h+2
=
h
√
= 4+h+2
7. Given f (x) = 3x2 − 3x + 5, evaluate the following expressions. Expand
and simplify when possible.
a. f (x + h)
f (x + h) = 3(x + h)2 − 3(x + h) + 5
= 3(x2 + 2xh + h2 ) − 3x − 3h + 5
= 3x2 + 6xh + 3h2 − 3x − 3h + 5
b. f (x) − f (x + h)
f (x) − f (x + h) = 3x2 − 3x + 5 − 3x2 + 6xh + 3h2 − 3x − 3h + 5
= 3x2 − 3x + 5 − 3x2 − 6xh − 3h2 + 3x + 3h − 5
= −6xh − 3h2 + 3h
c.
f (x)−f (x+h)
h
f (x) − f (x + h)
−6xh − 3h2 + 3h
=
h
h
=
h (−6x − 3h + 3)
h
= −6x − 3h + 3
4
8. A box with no top is to be created from a 16 x 25 piece of cardboard by
cutting out equal squares from all four corners and folding up the sides.
Write an expression for the volume of the box, V (x).
x
25 − 2x
x
x
16 − 2x
x
x
x
16
V = lwh = (16 − 2x)(25 − 2x)(x)
V (x) = 400x − 32x2 − 50x2 + 4x3
V (x) = 400x − 82x + 4x2
5
25
9. Derive the quadratic formula.
ax2 + bx + c = 0
By completing the squre, we can arrive at the Quadratic Formula.
b
c
x2 + x + = 0
a
a
b
c
x2 + x = −
a
a
2 2
b
b2
1 b
=
= 2
2 a
2a
4a
b
b2
b2
c
x2 + x + 2 = 2 −
a
4a
4a
a
2
2
b
b − 4ac
x+
=
2a
4a2
r
b2 − 4ac
b
=±
x+
2a
4a2
√
b2 − 4ac
b
x+
=±
2a
2a
√
2
b − 4ac
b
x=− ±
2a
2a
√
2
−b ± b − 4ac
x=
2a
6