Algebraic Manipulation Practice
College Algebra – Spring 2006
MAT 150
Name: ________________________
Week 2: Linear Equations and Linear Systems
Solving a linear equation.
Warm-up: Solve for x: a – (x + 1) = b + 1
1.
Solve for t: 22 1.3t  31.1
2.
Solve for x: 3 [5.4  2(4.3 x)]  2  (0.3x  0.8)
3.
Solve for x: ax  c  bx . Assume a  b (why?).
4.
Solve for q:
p2q  rq 1 4 p  r
Solving a linear system.
x  y  2
Warm-up: Solve: 
3x  y  6
1.
x

y 3

Solve, using the substitution method: 
2
 2 x  y  1 y

2.
 8x  5y  11
Solve, using the elimination method: 
2x  10y  1
3.
Find the (exact) point of intersection of the 2 lines:
y = x +1 and 2x + 3y = 12.
Solving a literal equation (or “formula”) for one of its variables.
1.
Solve S 
2.
Solve
rL  a
, for r; for a; for L.
r 1
1 1 1
  for R; then solve it for S.
R S T
MAT 150
Week 3: Quadratic Equations I
Name: ________________________
Solving a quadratic equation by factoring
Warm-up: Solve: 6 x 2  7 x  2  0
1.
Solve: 5 x( x  7)  0
2.
Solve: 2 x  3  8 x 2
3.
Solve: ( x  2)2  25  0
4.
Solve: 2 
1
x(9 x  12)
2
Solving a quadratic equation by taking square roots.
Warm-up: Solve: 3x 2  5  0
1.
Solve: 2( x  3)2  30  0
2.
Solve: 4 x 2  13  0
3.
Solve for x: ax 2  c  0 (Assume a  0 .)
Constructing a quadratic equation (with integer coefficients) from its roots.
In each of the following, find a quadratic equation (with integer coefficients) having the given
roots. Write your answer in the form: ax 2  bx  c  0.
Warm-up: Roots are: 2 and
6
7
1.
Roots are: -3 and 6
2.
Roots are:
1
and 2
3
3.
Roots are:
5
4
and
4
5
4.
Roots are: 1  3 and 1  3
MAT 150
Week 4 Quadratic Equations II
Name: ________________________
Solving a quadratic equation by using the quadratic formula x 
b  b2  4ac
2a
Warm-up: Solve, using the quadratic formula: 3x 2  5  4 x
1.
Solve: x 2  3  6 x  0
2.
Solve: 2 x 2  x  c  0
3.
Solve:
1 2
3
x  2 x   0 (Hint: Clear the fractions.)
3
2
Solving equations that are “quadratic in form.”
Warm-up: Solve: x 4  7 x 2  10  0 (The key: Making the correct change of variables.)
2
1.
 5 
 5 
Solve: 
  2
3  0
 x 1 
 x 1 
2.
Solve: 2 x 2 / 3  5 x1/ 3  3  0
3.
Solve: x 2  4 x 1  7  0
Finding the intercepts, axis of symmetry, vertex and range of a quadratic
function.
In each of the following, find the intercepts, axis of symmetry, vertex and range of the quadratic
function.
Warm-up: f ( x)  x 2  2 x  8
1.
f ( x)  x 2  6 x  9
2.
g ( x)   x 2  x  20
3.
h(t )  128t  16t 2
4.
f is the quadratic function whose vertex is at (2, 10) and whose graph passes through (6, 0).
MAT 150
Week 5: Exponents
Name: ________________________
Simplifying an expression using the laws of exponents
Warm-up: Simplify the following; have only positive exponents appear in your answer.

a 2 2ab3

2
6b 4
 
x 6 x3
1
1.
Simplify:
2.
Simplify: ( x1/ 3 y 2 )2  x 1 y 3 
3.
Simplify:
4 x 5
12u 3
3  uv 2 w4 
1
Simplifying a radical expression
Warm-up: Simplify the radical expressions completely:
(i) 3 16x 2 y 4
(ii)
9 x2  36
6x
(iii)
3
1.
Simplify:
3
2.
Simplify:
4
5 3
3.
Simplify:
5x
2 y3
4.
Multiply and simplify:
8x 2 y 4
 6  3  6  3 
4 xy 2
Rewriting a radical expression with rational exponents (and vice-versa).
Rewrite the radical expression, using exponents (or vice-versa).
Warm-up: (i)
1.
(iii)
 5xy 
2/3
8x y
2.
8x  y
3.
5  x  2
4.
(ii) 5xy1/ 3
7x3
3
2/3
5x
Solving an equation with rational exponents or radicals
Solve each of the following equations by hand.
Warm-up:
x 5 3  0
1.
Solve: 3 x  1  5
2.
Solve:
3.
Solve:
4.
Solve: 2 x  7  x  0
5.
Solve:
 2 x  1
1/ 3
4
3
5  0
x 1
4x 
2
1/ 3
2
MAT 150
Week 6: Exponential Equations
Name: ________________________
Solving equations of the form: bu  bv
Solve the exponential equations by first writing both members as powers of the same base.
Warm-up: Solve: 8 x  27  x
1.
Solve: 31 2 x  81
2.
Solve: 3x
3.
Solve: 9  3x 1  35 x
4.
Solve: 32 x  4  3x  3  0 . Hint: The equation is quadratic in form; use a u-substitution
2
2 x
 27
Using the Logarithmic Function: x  b y  y  l o gb x , to solve equations
having a single exponential expression. KEY: First isolate the exponential
expression.
Warm-ups: (a) Solve: 2 10 x1  5  83
1.
Solve: e3 x  5
2.
Solve: 5e0.04 x  1  16
3.
1
25  100  
2
(b) Solve for t in the formula: A  Pe rt .
t /6
(over)
Using the properties of logarithms (the product, quotient and power rules,
and the “change of base” formula) to solve exponential equations
Warm-up: Solve the following (by hand), giving exact solutions. Then, use your calculator to
compute these solutions to the nearest 0.001.
5  2e 2 x  207
Do the same for the following.
1.
2  3x  50
2.
115e0.06t  30
3.
53 x  2 x
4.
1500  850(1.085)t
5.
e2 x  7e x  10  0 . Hint: The equation is quadratic in form.
MAT 150
Week 7: Logarithmic Equations
Name: ________________________
Solving equations having only one logarithm
Solve the logarithmic equations.
KEY: First isolate
y
x  b  y  log b x to write the equation in exponential form.
the
logarithm;
then
use:
Warm-up: Solve: 2  3log( x  1)  9
1.
Solve: ln(2 x  7)  4  0
2.
Solve:
3.
I 
Solve for I in the formula: R  log  
 I0 
4.
Solve: log 3 ( x 2  2 x)  1 .
1
log 3 (7  x)  1
2
Solving equations having more than one logarithm
Solve the logarithmic equations. KEY: Use properties of logarithms (product, quotient and
power rules) to combine the logarithms into a single logarithm.
Warm-up: Solve: log x  log(8  x)  1
1.
Solve: log3 x  log3 ( x  6)  0
2.
Solve: 2 ln x  ln(8  2 x)
3.
Solve: log( x  3)  log( x  3)  2
4.
Solve: log4 (7  x)  log4 ( x  2)  0
(over)
Solving formulas containing exponential or logarithmic expressions
Warm-up: Solve the formula: B  Pe rt for t
Do the same for the following.
t
1.
 1 h
Solve the formula: N  N 0   for t
2
2.
 1 h
Solve the formula: N  N 0   for h
2
3.
Solve the formula: A  A0 e  kt for k
4.
L
Solve the formula: I  10log   for L
 L0 
5.
If Pekt  P(1  r )t , solve for r (in terms of k).
6.
If P  P0 e kt , solve the equation P  2 P0 for t.
t
MAT 150
Week 10: Completing the Square
Name: ________________________
Every quadratic equation can be solved by completing the square.
Warm-up: Solve by completing the square: 2 x 2  8 x  3  0
1.
Solve by completing the square: x 2  6 x  2  0
2.
Solve by completing the square: w2  5w  2  0
3.
Solve by completing the square: 3 x 2  4 x  3  0
Writing f ( x)  ax2  bx  c in the standard form f ( x)  a( x  h)2  k.
Warm-up: Write f ( x)  2 x 2  6 x  5 in standard form. Then, identify the axis of symmetry, the
vertex and the range of f.
Do the same for the following.
1.
f ( x)  x 2  4 x  7
2.
G ( x)   x 2  8 x  3
3.
h(t )  80t  16t 2
Finding the standard form for the equation of a circle: ( x  h)2  ( y  k )2  r 2 .
Warm-up: Write the equation of the circle: x 2  y 2  4 x  6 y  2  0 in standard form. Locate
its center and state its radius.
Do the same for the following.
1.
x2  y 2  6x  2 y  7  0
2.
x2  y 2  8x  7  0
3.
x2  y 2  5x  y  1  0
Rewriting a rational expression (by completing the square in its variable)
Rewrite each of the following by completing the square (in x).
Warm-up:
1
x  4 x  13
2
1.
Rewrite:
2
x  6 x  10
2.
Rewrite:
3
x  2ax
3.
Rewrite:
3
x  2 x  26
2
2
2
MAT 150
Week 11: Algebraic Fractions
Name: ________________________
Reducing an algebraic fraction
Simplify/reduce the following algebraic fractions. KEY: This can only be done by canceling the
FACTORS common to the numerator and denominator.
Warm-up:
x2  4 x  3
2 x 2  18
1.
x3  2 x 2  8 x
x2  x  2
2.
x2  2 x  1
x4 1
3.
8  2x2
x 2  5 x  14
4.
2x  6
Hint: The denominator is a sum of cubes and can be factored.
x 3  27
Multiplying and dividing algebraic fractions
Multiply or divide (and simplify). KEY: Factor all numerators and denominators, and divide by
multiplying by the reciprocal.
Warm-up:
a  b 8a  2

6 b2  a 2
1.
x2  x x  2

2x  4 x
2.
8 x 6 x3 1


y2
y xy
3.
x 2  3x
2x2  2

x2  x  6 x2  x  2
4.
x  x2  2 x  4 5x 



x2  4  x  2 x  2 
(over)
Adding and subtracting algebraic fractions
And or subtract (and simplify).
denominator).
Warm-up:
KEY: If necessary, first find the LCD (least common
3
5
2
2
x y
x
1.
y
2y  6
 2
y 3 y 9
2.
1
2
 2
a  3 a  4a  3
3.
x
2
3x  6

 2
x 1 x  2 x  x  2
Simplifying complex algebraic fractions
Simplify the following complex fractions. KEY: There are two methods for doing this.
Although either method can be used in any problem, practice will suggest which method will be
“easier” in a given problem.
Warm-up:
1
1 1

a b
1.
3
4

x x 1
2
3

x 1 x
2.
5
2

y 3 y
1
2

y y 3
3.
1  9 x 2
1  6 x 1  9 x 2
9
2x
5x2
1
4.
MAT 150
Week 12: Complex Number System
Name: ________________________
A complex number z is of the form z  a  bi , where a and b are real numbers and i  1 .
Terminology: a is called the real part and b is called the imaginary part of the complex
number z, and i is called the imaginary unit.
Complex Arithmetic: KEY: The arithmetic operations of addition, subtraction and
multiplication of complex numbers have been defined so that these operations can be carried out
by treating the complex numbers as if they were binomials.
Warm-ups: Perform the indicated operations and simplify (writing your answer in a  bi form).
(a) (3  4i)  (9  i)
(b) (7  2i)  (3  5i)
(c) (4  i)(3  2i)
Exercises:
1. If z1  4  3i and z2  1  5i , find the following (and simplify).
a. z1  z2
b. z1  z2
c. z1  z2
2. If z1  3  2i and z2  5i , find the following (and simplify).
a. z1  z2
b. z1  z2
c. ( z1 )2
3. If z1  5  7i and z2  5  7i , find the following (and simplify).
a. z1  z2
b. z1  z2
c. ( z1 )2
(Principal) Square Root of a Negative Number: For any N  0 , we define
 N  i N . KEY: When working with such radicals, first express the number in terms of the
imaginary unit, i.
Warm-ups: Perform the indicated operations and simplify (writing your answer in a  bi form).
(a)
18 2
(b)

16 5  9
Do the same in each of the following:
4.
7 7
(over)


(c) 6  3

2

5. 8  4

2
6. 5 4  3 9
The Division of Complex Numbers: This last arithmetic operation is done by making use
of the complex conjugate. Definition: If z  a  bi , then its complex conjugate is z  a  bi .
The product of a complex number with its conjugate always produces a nonnegative real
number.
Warm-up: Divide (and simplify):
5  2i
4  3i
Do the same in each of the following:
7.
4  3i
5  2i
8.
3i
4i
9.
1 i
6  2i
Solve the following equations. (Write any complex solutions in a  bi form.)
10. x 2  4 x  1  0
11. x 3  8  0 (Note: This equation has 3 solutions.)
12. x3  x 2  4 x  4  0
MAT 150
Week 13: Division of Polynomials
Name: ________________________
Let f(x) and g(x) be polynomial functions, where the degree of f  the degree of g. Then, there
are unique polynomials q(x) and r(x), such that
f ( x)
r ( x)
and, hence, f ( x)  q( x) g ( x)  r ( x).
 q ( x) 
g ( x)
g ( x)
(1)
In (1), f(x) is called the dividend, g(x) is called the divisor, q(x) is called the quotient, and r(x)
is called the remainder (which may be a constant, possibly 0). If the remainder r ( x)  0, we
say that the divisor g(x) is a factor of the dividend f(x).
Long Division: A comprehensive way of finding q(x) and r(x)
Warm-ups: Perform the following division (use long division); write your answer in the form
given in (1).
x3  2 x 2  3x  5
a.
x 1
x3  2 x 2  5
c.
x2  x  2
x3  3x  2
b.
x2
Exercises: Do the same for the following.
1.
x2  2 x  5
x3
2.
x3  2 x 2  5 x  2
x 1
x3  27
3.
x3
4.
x3  a3
xa
5.
x4  5x2  6
x2  4
6.
2 x 2  3x  5
x2  x  2
(over)
Synthetic Division: When the divisor g can be written in the form: g ( x)  x  k , k a
constant, then an alternative method can be used to find
Warm-ups: Use synthetic division to find the following.
a.
x2  4 x  3
x2
b.
Do the same in each of the following:
7.
x2  5x  8
x 3
8.
x3  2 x  x  4
x 1
9.
x2  5x  8
x 3
x2  6x  5
10.
1
x
2
11.
x3  8
x2
x3  5 x  1
x3
f ( x)
.
g ( x)
MAT 150
Week 14: Difference Quotient
Name: ________________________
Definition: Let f be a function and let x and x + h be in its domain. Then
f ( x  h)  f ( x )
h
is called a difference quotient of f. The difference quotient has two fundamental interpretations:
(1) It gives the slope of the secant line that joins the points P = (x, f(x)) and Q = (x + h, f(x + h))
on the graph of y = f(x), and (2) it equals the average rate of change of the value of f(x) on the
interval [x, x + h]. A central idea in the Calculus (viz., the derivative) utilizes the concept of the
difference quotient.
Finding the Difference Quotient: Find (and simplify) the difference quotient for each of
the following.
Warm-ups:
(a) f ( x)  2 x 2  3x  5
(b) g ( x) 
Do the same for each of the following:
1. f ( x)  8 x  3
2. f ( x)  x 2  6 x
3. g ( x)  x3
4. s(t )  144t  16t 2
5. f ( x ) 
1
2x
6. F ( x ) 
8
x2
(over)
4
x 1
(c) F ( x)  8 x
7. S ( x)  x
8. f ( x)  mx  b
9. f ( x)  ax 2  bx  c
10. G ( x) 
1
x3
11. j ( x)  3 4  x
12. g ( x)  f ( x)  k (k a constant) How do the difference quotients of f and g compare?
13. g ( x)  cf ( x) (c a constant) How do the difference quotients of f and g compare?
MAT 150
Week 15: Summation Notation
Name: ________________________
Summation notation is shorthand for writing lengthy sums:
n
a
i 1
i
 a1  a2  a3  . . .  an
(1)
In (1), i is called the index of summation, and 1 and n are called the lower and upper limits of
summation, respectively. an is called the general, or nth term of the sum.
(Re)writing a sum using summation notation
Rewrite the following sums using summation notion. Key: Look for patterns (e.g. successive
terms differ by the same amount or the form a common ratio, or the terms alternate in sign, etc.).
Use this to write a formula for the general term, an .
Warm-ups:
(a) 2  5  8  . . .  (3n  1)
(b) 3  5  7  . . .  77
1. 3  6  9  . . .  90
2.
1 1 1
  . . .
2 4 8
3. 5  9  13  17  . . .  101
4. 1  4  9  16  . . .
5. (1  b)  (1  2b)2  (1  3b)3  . . .  (1  13b)13
(over)
(c) 5  10  20  40  80  . . .
Writing out the terms of a given sum
Write out the terms of the following sums and identify how many terms there are in the given
sum. (When there are many terms, use ellipses. . . .)
Warm-ups: (a)
100
 (2k  3)
k 1
6.
(b)
50
  5i  2 
i 8
(c)

2n

n
n 0 5  1
20
7
i1
8
7.
 (1)
n 1
5
n 1
8.
n
 ar
k 1
k 1
9.
n
  a  (k  1)d 
k 1

10.
1
1
 n!
Hint: Recall that n! (n-factorial) is defined as
n 0
1,
n!  
n   n  1   n  2  
27
11.
 k (k  1)
k 1
n
12.
 f ( x )x , where f is some function of x.
i 1
i
if n  0
 3  2 1, if 1  n
.
MAT 150
Name: ________________________
Week 16: Absolute-Value Equations/Inequalities
KEY: The following “geometric interpretation” of absolute-value is, in my opinion, most
helpful for understanding how absolute-value equations and inequalities can be solved by hand.
Consider: State the values of: 7 
, 5 
, and 0 
. Note that (and here’s the
key), for any real number x,
x = the distance of the number x from the origin (0) in the number line.
Solving Absolute-Value Equations: Solve the following:
Warm-ups:
(a) x  6
Do the same for each of the following:
1. x  1  9
2. 7  x  5
3.
x2
1  7
3
4. x  5  x  3
5. 1  2x  x
6.
x
7 5 3
3
(over)
(b) 2 x  5  8
(c)
2x  3
3 7
5
Solving Absolute-Value Inequalities: Solve the following and graph the solution set, S,
on a number line.
Warm-ups:
(a) x  5
(b) x  5  3
(c) x  7
(d) 4  3x  10
Do the same for each of the following:
7. 2 x  1  7
8. 8  x  4
9.
x 1
2
7
10. 3x  10  10
Problem: Find all those numbers x on the number line that are twice as far from 2 as they are
from 10.