College Algebra Tutorial

I. Concepts of Algebra
1. Exponents and Radicals
2. Factoring
3. Fractional Expressions
4. Graphical Representation of Data
II. Equations and Inequalities
1. Linear equations
2. Quadratic Equations
3. Complex Numbers
4. Other Types of Equations
5. Linear Inequalities
6. Other Types of Inequalities
III. Functions and Graphs
1. Lines
2. Function Concepts
3. Functions and Graphs
4. Shifting Graphs
5. Combining Functions
6. Inverse Functions
IV. Exponential and Logarithmic Functions
1. Exponential Functions
2. Logarithmic Functions
3. Laws of Logarithm
4. Exponential and Logarithmic Equations

Exponents and Radicals

I. Simplify each of the
following expressions.

II Radicals

1.

7. Find the value of

2.

8. Simplify by removing all possible factors from the radical.

3.

9. Simplify by removing all possible factors from the radical.

4.

10. Simplify by removing all possible factors from the radical.

5.

11. Rationalize the denominator and simplify.

6.

12. Rationalize the denominator and simplify.

Simplify the expression

Commute the terms so as to group the constant terms together

and
group the variable terms together.

Simplify by using the first law of exponents.

Simplify the expression

x3(2y z2)3

x3(2y z2)3

= x3 . 23 y3 (z2)3
= x3 . 8y3z6
= 8 x3 y3 z6

Distribute the exponent of 3 among the three factors

in (2y z2)3 using the properties of exponents.

Simplify by using the properties of exponents and

write your answer in standard form.

Simplify the expression

Use the commutative law to rearrange the factors of the

numerator and simplify the denominator using one of the
properties of exponents.

Simplify the numerator by multiplying the constant term and

combining the variable terms using the properties of
exponents.
Combine the variable terms found in the numerator and
denominator using the properties of exponents.
Simplify the expression

Simplify the second factor in the numerator.

Simplify the numerator further by combining the variable

terms
and the constant terms.
Combine the variable terms found in the numerator and

denominator.
Write this expression without negative exponents.
Simplify the expression

Distribute the -3 exponent among the three factors in

parentheses using the properties of exponents.
Simplify each of the three terms using the properties of
exponents.
Simplify and write this expression with no negative
exponents.
Simplify the expression

Distribute the fractional exponent among the factors in

parentheses using properties of exponents.
Simplify each of these three factors using properties of
exponents
Continue to simplify each factor.
Write this expression without negative exponents.

Find

Try to think of a number whose cube is 64

Simplify by removing all possible factors from the radical

Using properties of radicals, rewrite the single radical as a

product of three radicals.

Simplify each radical where possible.

Simplify by removing all possible factors from the radical

Rewrite each of the 3 factors under the radical as a product

so that you can take the cube root of one of the factors.

Rewrite the expression as a product of radicals.

Find the cube root of each factor where possible.
Simplify by removing all possible factors from the radical

Write the expression as a quotient of two radicals, and

factor the numerator in preparation to extract a radical.
Write the numerator as a product of two radicals.
Simplify by extracting square roots where possible.
Rationalize the denominator and simplify

You are to write the given fraction as an equivalent fraction

where the new fraction has no radical in the denominator.

Simplify the denominator.

Rationalize the denominator and simplify

Multiply the numerator and denominator by the

conjugate of the denominator.
Multiply the two binomials in the denominator.
i.e. FOIL the denominator.

Simplify the denominator.

Continue simplifying the denominator.
Reduce the constant coefficients in this fractional
expression.
Fractional Expressions
Simplify the rational expressions
and write them in reduced form.

10.

1.

11.

2.

12.

3.

13.

4.

14.
Perform the operation and
write the result in reduced form.

15.

5.

16.

6.

17.

7.

18.

8.

19.

9.

Simplify the rational expression

and write it in reduced form.

Factor a common factor

from
the two terms in the
denominator.
Divide a common factor
from the numerator and
denominator.
Write the expression in
reduced form.
Simplify the rational expression
and write it in reduced form.

Factor the numerator. Note that the numerator

has four terms. This should suggest a
factoring
method.
Continue factoring the numerator by
grouping.
Note that 2 - a is the negative of a - 2.
Reduce this fraction by dividing a - 2 into the
numerator and into the denominator.
Write this rational expression in simplified
reduced form.
Simplify the rational expression
and write it in reduced form.

Notice that the numerator is a sum

of two cubes. Write it as such.
Factor the numerator using the formula
for factoring the sum of two cubes.
Divide out the common factor found in the
numerator and in the denominator.
= (a + 2)

Write this rational expression in simplified

reduced form.

Perform the multiplication

and write the results in reduced form.

Perform the operation by multiplying

the numerators together and
multiplying the denominators together.
Factor the difference of two squares which
is found in the denominator.
Divide out the common factor found in the
numerator and in the denominator.
Continue to reduce this fractional
expression by
dividing the numerator and denominator
by a.
Write this rational expression
in simplified reduced form.
Perform the multiplication
and write the results in reduced form.

Perform the operation by multiplying

the numerators together and
multiplying the denominators together.
Completely factor the numerator and
completely factor the denominator
Divide out the common factors found in the
numerator and in the denominator.
= (a + 3 )

Write the expression

in simplified reduced form.

Perform the multiplication

and write the results in reduced form.

Perform the operation by

multiplying the numerators
together and multiplying the
denominators together.
Factor the numerator and
denominator.
Notice that the numerator contains
a factor which is a difference of
two cubes. Write this factor as a
difference of two cubes.
Factor the difference of two cubes
found in the numerator.

Divide out terms common to the

numerator and denominator.

Perform the multiplication

and write the results in reduced form.

Perform the operation by multiplying the numerators

together and multiplying the denominators together.
Factor the numerator and denominator.
Divide out terms common to the numerator and
denominator.
Write the expression in reduced form.
Perform the division
and write the results in reduced form.

Perform the operation by

multiplying the numerator by the
reciprocal of the denominator.
Factor the numerator and
denominator. Note the numerator
contains a difference of two
squares.
Factor the numerator further.
Divide out terms common to the
numerator and denominator.
Write the expression in reduced
form.
Perform the division
and write the results in reduced form.

Perform the operation by multiplying the

numerator by the reciprocal of the denominator.
Factor the numerator and denominator. Note the
denominator contains a difference of two cubes
and a sum of two cubes.

Factor the denominator further.

Divide out terms common to the numerator and
denominator.
Write the expression in reduced form.
Perform the addition
and write the results in reduced form.

Factor the denominators into prime factors so as to assist in

finding the least common denominator.
2.52.3 = 150

What is the least common denominator?

Write each fraction in an equivalent form with 150 as the
common denominator.
Perform the addition.
Perform the subtraction
and write the results in reduced form.

(2x + 1)2

Find the "least" common denominator for these two

fractions.
Write each fraction in an equivalent form with
(2x + 1)2 as the common denominator.
Perform the subtraction
Simplify the expression.
Perform the subtraction

and write the results in reduced form.

(x + 2)

Find the "least" common denominator for these two

fractions.
Write each fraction in an equivalent form with
(x + 2) as the common denominator.
Perform the subtraction
Simplify the expression.
Perform the addition
and write the results in reduced form.

Notice that (2 - x) = - (x - 2). Rewrite the denominator of the

second fraction using this equivalence.
Write each fraction in an equivalent form with
(x - 2) as the common denominator.
Perform the addition.
Simplify the expression.
Perform the addition
and write the results in reduced form.

Factor each denominator to assist in

finding the "least" common denominator.
(x - 1)(x - 2)(x + 2)

Using the information above, what is the

"least" common denominator?
Write each fraction in an equivalent form
with (x - 1)(x - 2)(x + 2) as the common
denominator.
Perform the subtraction.
Simplify the expression.

Perform the addition

and write the results in reduced form.

Factor each denominator to assist

in finding the "least" common
denominator.
(y + 2)2 (y-1)

Using the information above,

find the "least" common
denominator.
Write each fraction in an
equivalent form with
(y + 2)2(y-1) as the common
denominator.
Perform addition.
Simplify the expression.

Perform the operations

and write the result in reduced form.

The two terms of the numerator first need to be combined

into a single fractional expression. What is the least

common denominator of the two terms in the numerator?

Write each fraction in the numerator as an equivalent
fraction with 3 as the common denominator.
Combine the two terms of the numerator into a single
fraction.
Multiply the numerator and denominator by the reciprocal
of the denominator.
Simplify this expression.

Perform the operation

and write the result in reduced form.

y ( y + 1)

The two terms of the numerator first need to be combined

into a single fractional expression. What is the least
common denominator of the two terms in the numerator?
Write each fraction in the numerator as an equivalent
fraction with y(y + 1)as the common denominator.
Combine the two terms of the numerator into a single
fraction.
Simplify the fraction in the numerator.
Multiply the numerator and denominator by the reciprocal
of the denominator.
Simplify this expression.

Graphical Representation of Data

1. Determine the quadrant in which (x , y) is located so that x < 0 and y > 0.

2. Determine the quadrant in which (x , y) is located so that - x < 0 and y < 0.
3. Determine the quadrants in which (x , y) is located so that x < -4 .
4. a) Sketch the line segment with endpoints at (-2,3) to (4,7).
b) Find the midpoint of the segment with endpoints at (-2,3) to (4,7).
c) Find the length of segment with endpoints at (-2,3) to (4,7).

5. a) Plot the triangle with vertices at A( 8, 5 ), B( 1, -2 ) and C( -2, 2 ).

b) Show this is a right triangle by :
i) finding the length of each of the three sides and
ii) showing that these three lengths satisfies the Pythagorean Theorem.
Graphical Representation of Data

1. Determine the quadrant in which (x , y) is located so that x < 0 and y > 0.

Sketch the part of the (x,y) plane where x < 0.

Sketch the part of the plane where y > 0.

The intersection of the above two shaded areas

will give the part of the plane where the x < 0
and y > 0. Find this intersection by first overlaying
the two shaded areas.
Quadrant II

The part of the plance where x < 0 and y > 0 is the

crossed hatched area. Which quadrant is this?
Graphical Representation of Data

2. Determine the quadrant in which (x , y) is located so that - x < 0 and y < 0.

(-1)(-x) > (-1) 0
x>0

In order to more easily picture the part of the plane

where -x < 0, multiply both sides of this inequality
by -1. Remember to switch the direction of the
inequality

Sketch the part of the plane where x > 0.

Sketch the part of the plane where y < 0

The intersection of the above two shaded areas

will give the part of the plane where the x > 0
and y < 0. Find this intersection by first overlaying
the two shaded areas.
Quadrant IV

The part of the plance where x > 0 and y < 0 is the

crossed hatched area. Which quadrant is this?
Graphical Representation of Data

3. Determine the quadrants in which (x , y) is located so that x < -4 .

Sketch the part of the (x,y) plane where x < -4.

The points (x,y) where x < -4

lie in either quadrant II or III.

In which quadrants do the points (x,y) lie so that

x < -4.
Graphical Representation of Data

4. a) Sketch the line segment with endpoints at (-2,3) to (4,7).

b) Find the midpoint of the segment with endpoints at (-2,3) to (4,7).
c) Find the length of segment with endpoints at (-2,3) to (4,7).

Sketch the line segment with endpoints at (-2,3) to (4,7).

Recall the midpoint formula for a line segment.

Substitute into the midpoint formula in order to find
the midpoint of the segment from (-2,3) to (5,7). Write
your answer in simplified form.

Place this midpoint (1,5) on the segment and check that

it visually looks like the midpoint of the segment.

Recall the formula for the distance between two points.

Substitute into this formula to find the distance
from (-2,3) to (5,7).
Graphical Representation of Data

5. a) Plot the triangle with vertices at A( 8, 5 ), B( 1, -2 ) and C( -2, 2 ).

b) Show this is a right triangle by :
i) finding the length of each of the three sides and
ii) showing that these three lengths satisfies the Pythagorean Theorem.

Plot the three points A( 8, 5 ), B( 1, -2 ) and C( -3, 2 ).

Find the length of side ABby using the distance formula.

Find the lengths of side BC and CA.

Check whether the lenghts satisfies the criteria for the

Pythagorean Theorem, i.e. check whether
AB2 + BC2 = AC2.
Linear Equations
1. Determine which of the x values are solutions of the equation x4 + x3 - 5x2 + x - 6 = 0.
a) x = -3, b) x = -2, c) x = 1, d) x = 2
2. Determine whether the equation - 5(x+2) +2 = -3x + 5 is true for all values of x (and hence an
identity) or if is true for only some values of x (and hence a conditional equation.)
3. Determine whether the equation - 4(x-3) + 7x = 3(x+4) is true for all values of x (and hence an
identity) or if is true for only some values of x (and hence a conditional equation.)
Solve the equation if possible and check your solution:
4. 7x - 1 = 3(x + 5)
5. 5[x - (3x + 2)] = 9 - 5x
6. 1.4x - .8 = .3(4 - 5x)

7.
8.

9.
10.
11.
12.
Solve for x:
13. 5(x+2) + ax = 5 - x
14. 6 - bx = 15 + cx
15.

Determine which of the x values are solutions to the equation

a) x = -3, b) x = -2, c) x = 1 , d) x = 2.

Substitute -3 into the equation for x and determine if the equation is

satisfied.
Evaluate this expression and determine if the two sides are equal.
Yes, since both sides are 0. Is x = -3 a solution?
Substitute -2 into the equation for x and determine if the equation is
satisfied.
Evaluate this expression and determine if the two sides are equal.
No, since the two sides are
Is x = -2 a solution?
unequal.
Substitute 1 into the equation for x and determine if the equation is
satisfied.
Evaluate this expression and determine if the two sides are equal.
No, since the two sides are
Is x = 1 a solution?
unequal.
Substitute 2 into the equation for x and determine if the equation is
satisfied.
Evaluate this expression and determine if the two sides are equal.
Yes, since both sides are 0 Is x = 2 a solution?
Determine whether the equation
- 5 (x + 2 ) + 2 = -3x + 5
is true for all values of x (and hence an identity) or if it is true for only some values of x (and
hence a conditional equation.)

- 5 (x + 2 ) + 2 = -5x -10 + 2
= -5x - 8

Simplify the left hand side and determine if it is

identical to the right hand side

No, because the simplified

left side is not equal to the
right hand side.

Is this an identity?

- 5 (x + 2 ) + 2 = -3x + 5

Find the values for which this conditional

equation is true.

-5x - 8 = -3x + 5

Simplify the left hand side.

-5x + 3x = 5 + 8

Perform operations to write an equivalent

equation with the variables on the left and the
constants on the right.

-2x = 13
x = - (13 / 2)

Simplify and solve.

Determine whether the equation
-4 (x-3) + 7x = 3(x+4)

is true for all values of x (and hence an identity) or if it is true for only some values of x (and
hence a conditional equation.)

-4 (x-3) + 7x = -4x + 12 +7x

= 3x + 12
= 3(x + 4)

Simplify the left hand side and determine if it is

identical to the right hand side

Yes, because the left side is

shown to be equal to the
Is this an identity?
right hand side.
Solve the equation
7x - 1 = 3 (x + 5)
and check your answer

7x - 1 = 3 (x + 5)
Apply the distributive law to the right hand side.
7x - 1 = 3x + 15
7x - 3x = 15 + 1

Transform the equation to one with the variables on the left and the
constants on the right.

4x = 16
Simplify each side and solve for x.
x=4
7(4) - 1 = ? 3(4 + 5)
28 -1 ?= 3(9)
Check your answer by substituting x = 4 into the equation.
27 = 27
Yes, x = 4 is a solution.
Solve the equation
5 [x - (3x + 2)] = 9 - 5x
and check your answer

5 [x - (3x + 2)] = 9 - 5x
5[x -3x-2] = 9 - 5x
Simplify the left hand side.
5[-2x-2] = 9 - 5x
-10x -10 = 9 - 5x
-10x + 5x = 9 + 10

Transform the equation to one with the variables on the left and
the constants on the right.

-5x = 19
Simplify each side and solve for x.
x = -(19 / 5)

- 19 + 3 (19) - 10 =? 9 + 19 Check your answer by substituting x = -(19 / 5) into the equation.

28 = 28
Yes, x = -(19 / 5)is a solution.

Solve the equation

1.4x - .8 = .3(4 - 5x)

and check your answer

1.4x - .8 = .3(4 - 5x)

Apply the distributive law to the right hand side.
1.4x - .8 = 1.2 - 1.5x
(10) [1.4x - .8 ] = (10) [1.2 Multiply both sides by 10 so as to create an equivalent equation
1.5x]
with no decimals in it.
14 x - 8 = 12 - 15x
14x + 15x = 12 + 8
29x = 20

Transform the equation to one with the variables on the left and the
constants on the right.
Simplify each side and solve for x.

Check your answer by substituting

Solve the equation

Multiply both sides by the least common denominator to create an

equivalent equation with no denominators.
10 - 15 x = 2x Simplify both sides.
Solve for x.

Solve the equation

Multiply both sides by the least common denominator to create an

equivalent equation with no denominators.
Simplify both sides.
12x + 8 = 6(x + 5)
Continue simplifying both sides.
12x + 8 = 6x + 30
Transform the equation to one with the variables on the left and the
12x - 6x = 30 - 8 constants on the right.
6x = 22

Simplify each side and solve for x.

Solve the equation

Factor the denominator on the right hand

side so as to help in determining a "least"

common denominator.
Multiply both sides by the "least" common
denominator to create an equivalent equation
with no denominators.
Apply the distributive law on the left side
and perform the multiplication on the right.
3(x-3) + (x+2) = 4 Reduce each fractional expression.
3x - 9 + x + 2 = 4
4x - 7 = 4
4x = 11 Simplify each side and solve for x.
Since x = 11 / 4 is not a zero of any
of the denominators, it is in fact a valid solution.

Is this answer an extraneous solution or a

valid solution?

Solve the equation

Factor the denominator on the left hand side so as

to help in determining a "least" common
denominator.
Multiply both sides by the "least" common
denominator to create an equivalent equation with
no denominators.
Apply the distributive law on the left side and
perform the multiplication on the right.
7 - 4(x - 2) = 5 (x + 2) Reduce each fractional expression.
7 - 4x + 8 = 5x + 10
15 - 4x = 5x + 10
-9x = -5 Simplify each side and solve for x.
Since x = 5 / 9 is not a zero of any
of the denominators, it is in fact a valid
solution.

Is this answer an extraneous solution or a valid

solution?

Solve the equation

Multiply both sides by the "least" common denominator to

create an equivalent equation with no denominators.
Apply the distributive law on the right side and perform the
multiplication on the left.
9y = 3 + 2(3y - 1) Reduce each fractional expression.
9y = 3 + 6y - 2
3y = 1 Simplify each side and solve for y.
Since x = 1/3 is a zero of two of Is this answer an extraneous solution or a valid solution?
the denominators, these

fractions are undefined at x =

1
/3 . Hence this is an extraneous
solution.
Solve the equation

Factor the denominator on the right. hand side

so as to help in determining a "least" common
denominator.
Multiply both sides by the "least" common
denominator to create an equivalent equation
with no denominators.
Apply the distributive law on the left side and
perform the multiplication on the right.
y + 2 + 4 (y - 2) = 5y - 6 Reduce each fractional expression.
y + 2 + 4y - 8 = 5y - 6
Simplify each side and solve for y.
5y - 6 = 5y - 6
Since this equation is an identity, it is true for
all values of y. However, in determining the
We note that the above equation is an identity.
solution to the original equation, we must
How do we interpret this fact with regard to the
exclude all values of y where the denominators
solution set to the original equation?
are zero. Hence, the solution set is all values of
y except y = 2 and y = -2.
Solve the equation

for x.

Strategy: Transform the equation to one with the variable terms on

one side and the constant terms on the other. Factor out x from the
side containing it and then solve for x.
5x + 10 + ax = 5 - x Apply the distributive law on the left hand side.
5x + ax + x = 5 - 10

Write an equivalent equation with all variables on the left and the
constant terms on the right.

6x + ax = - 5
(6 + a) x = - 5 Simplify, factor out an x and then solve for x.

Solve the equation

for x.

Strategy: Transform the equation to one with the variable terms on

one side and the constant terms on the other. Factor out x from the

side containing it and then solve for x.

6 - 15 = cx + bx

Write an equivalent equation with all variables on the right and the
constant terms on the left.

-9 = (c + b) x
Simplify, factor out an x and then solve for x.

Solve the equation

for x.
Strategy:
1. Rewrite the equation as an equation with no denominator.
2. Transform the equation to one with the terms involving x
on one side and all remaining terms on the other side.
Factor out x from the side containing it and then solve for x.
y(2-3x) = 2x

Rewrite the equation as an equation with no denominator

by multiplying both sides by 2 - 3x.

2y - 3yx = 2x Expand the left hand side by using the distributive law.
Write an equivalent equation with terms involving x on the
2y = 2x + 3yx right
and all remaining terms on the left.
2y = (2 + 3y) x
Factor out an x and then solve for x.
Quadratic Equations
Write the quadratic equation in standard form.
1. 6x = 5 - 2x2
2. 9 = 2(2x +4)2
Solve the equation by factoring.
3. 2y2 - 5y -3 = 0
4. 12 - 15x2 = 8x

5.
6. y2 - 8ay - 9a2 = 0
Solve the equation by extracting square roots.
7. 4y2 = 25
8. (y - 3)2 = 10
Solve the equation by completing the square.
9. y2 + 6y + 7 = 0

10. 9y2 - 36y = -5

Solve the equation by using the quadratic formula.
11. 3x2 + 2x -1 = 0
12. 4y - 2 = y2

13.
Write the quadratic equation in standard form.
6x = 5 - 2x2
Rewrite the equation with all terms on one side of the equal sign,
6x = 5 - 2x2
with the x2 term first, with the x term in the middle and the
2x2 + 6x - 5 = 0
constant term last
Write the quadratic equation
9 = 2(2x + 4)2

in standard form.
9 = 2(2x + 4)2

9 = 2(4x + 16x + 16) Expand the expression on the right hand side of the equation.
9 = 8x2 + 32x + 32
2

Rewrite the equation with all terms on one side of the equal sign,
0 = 8x2 + 32x + 23 or
with the x2 term first, with the x term in the middle and the
8x2 + 32x + 23 = 0
constant term last.
Solve the equation
2y2 - 5y - 3 = 0
by factoring.
2y2 - 5y - 3 = 0 Factor using the usual factoring method for
(2y + 1)(y - 3) = 0 trinomials.
2y + 1 = 0 ; y - 3 = 0

This product will be zero if either (or both) factors

are zero. Set each factor equal to zero.

y = - (1/2) ; y = 3 Solve each of these two linear equations.

Solve the equation
12 - 15x2 = 8x
by factoring.
12 - 15x2 = 8x
0 = 15x2 + 8x -12 or Write the equation in standard form.
15x2 + 8x -12 = 0
(5x+ 6)(3x - 2) = 0
5x+ 6 = 0 ; 3x - 2 = 0

Factor using the usual factoring method for

trinomials.
This product will be zero if either (or both) factors
are zero. Set each factor equal to zero.

x= - ( 6/5 ) ; x = 2/3 Solve each of these two linear equations.

Solve the equation

by factoring.
Strategy: 1. Factor the denominator.
2. Multiply both sides by the least common
denominator.
3. Write the equation in standard quadratic
form.
4. Factor and solve the equation.
Factor the denominator on the right hand
side.
Multiply both sides by the "least" common
denominator.
Use the distributive law and perform the
multiplication.
2x2 + 5(x + 3) - 4x2 - 12x = 18
2x2 + 5x + 15 - 4x2 - 12x = 18 Simplify each fractional expression and then
-2x2 -7x - 3 = 0 write the equation in standard quadratic form.
2x2 + 7x + 3 = 0
(2x + 1)(x + 3) = 0 Factor the equation.
2x + 1 = 0; x + 3 = 0 Set each factor equal to zero and then solve
x = - (1/2) ; x = - 3 each of these two linear equations.
Solve the equation
y2 - 8ay - 9a2 = 0
for y by factoring.
(y - 9a)(y + a) = 0 Factor using the usual factoring method for
trinomials.
y - 9a = 0 ; y + a = 0

This product will be zero if either (or both) factors

are zero. Set each factor equal to zero.

y = 9a ; y = -a Solve each of these two linear equations for y.

Solve the equation
4y

2 = 25

by extracting square roots.

Divide both sides by 4.
Apply the square root operator to both sides and
simplify..
Solve the equation
(y - 3)2 = 10
for y by extracting square roots.

Apply the square root operator to both sides the

equation.
Solve for y .
Solve the equation
y2 + 6y + 7 = 0
by completing the square.
Strategy: 1. Rewrite the equation with just the variable
terms on the left.
2. Complete the square on the left side.
3. Extract square roots from both sides.
4.Solve the equation.
y2 + 6y = -7
y2 + 6y + ___= -7 + ___

Rewrite the equation with just the variable terms

on the left.
Add the same number to both sides so as to make
the left side a perfect square.

This is done by taking half the coefficient of the

y + 6y + 3 = -7 + 3 linear term (i.e. half of 6), squaring that number
and adding it to both sides.
2

(y + 3) 2 = -7 + 9 Write the left side as a perfect square and simplify

(y + 3)2 = 2 the right side.
Extract square roots from both sides and solve for
y.
Solve the equation
9y2 - 36y = -5
by completing the square.
Strategy: 1. Rewrite the equation with just the variable
terms on the left.
2. Complete the square on the left side.
3. Extract square roots from both sides.
4.Solve the equation.
Rewrite the equation with just the variable terms
9(y2 - 4y) = -5
on the left. Add the same number to both sides so
9(y2 - 4y + ___) = -5 + ___
as to make the left side a perfect square.
This is done by taking half the coefficient of the
linear term (i.e. half of 4), squaring that number,
9(y2 - 4y + 22 ) = -5 + 9.22
multiply it by the coefficient 9 and then adding
that product to both sides.
9(y - 2) 2 = -5 + 36
9(y - 2) 2 = 31

Write the quadratic polynomial on the left side as

a perfect square and simplify the right side.
Simplify the right hand side.
Divide both sides by 9.
Extract square roots from both sides.
Solve for y.

Solve the equation

using the quadratic formula.

5y = -1(y2 -9)
Rewrite this equation without a denominator
5y = -y2 + 9
y2 + 5y - 9 = 0 Write the equation in standard form.
Recall the quadratic formula.
a = 1; b = 5 c = -9 Write the values of a, b, c.
Substitute the values of a, b, c, into the quadratic
formula.
Simplify this expression.
Express the two solutions in simplified form.
Complex Numbers
I. Write the complex number in standard form (i.e. in the form a + b i.)
1.

2. -5i + 3 i 2
II. Perform the operation and write the result in standard form.
3. (2 + 3i) - (6 -4i)
4.

5. (2 + 3i)(1 - 5i)
6. 7i(7 - 3i)
7. (3 + 2i)2 + (4 - 3i)
III. Write the conjugate of the complex number.
8. 3 + 5i
9. 2 - 6i
10. 15i
IV. Perform the operation and write the result in standard form.
11.

12.

13.

V. Solve the quadratic equations using the Quadratic Formula.

14. 3x2 + 9x +7 = 0
15. y2 -2y + 2 = 0
Write the complex number

in standard form

Write this expression in

standard form.
Write the complex number
-5i + 3 i 2
in standard form
Write i 2
as a real
number.

-5i + 3 i = -5i + 3 (-1)

Write this
expression
in
standard
form.

= -3 - 5i

Perform the operation and

(2+3i) - (6 - 4i)
write the result in standard form
(2 + 3i) - (6 - 4i) = 2 + 3i - 6 + 4i
= 2 - 6 + 3i + 4i
= -4 + 7i

Rewrite the expression without

parentheses.
Combine the real parts and combine the
imaginary parts.
Write the result in standard form.

Perform the operation and

write the result in standard form

Simplify the radicals and express
them using complex notation.
Combine the real parts and combine
the imaginary parts.
Write the result in standard form.
Perform the operation and

(2 + 3i) (1 - 5i)
write the result in standard form
(2 + 3i) (1 - 5i)

= 2 - 10 i + 3i - 15 i 2
= 2 - 10 i + 3i - 15(-1)

Expand the product, and simplify.

= 2 + 15 - 10 i + 3i
= 17 - 7i

Combine the real parts and combine the

imaginary parts.
Write the result in standard form.

Perform the operation and

7i (7 - 3i)
write the result in standard form
7i (7 - 3i)

= 49i - 21i2
= 49i - 21(-1)

Expand the product and

simplify.

= 21 + 49 i

Write the result in standard

form.

Perform the operation and

(3 + 2i) 2 + (4 - 3i)
write the result in standard form
(3 + 2i) 2 + (4 - 3i)

= (9 + 6i + 6i + 4i2) + (4 - 3i)
= (9 + 12i + 4(-1)) + (4 - 3i)

Expand the squared term and simplify

= 9 -4 + 4 +12i - 3i
= 9 + 9i

Group the imaginary terms together,

group the real terms together.
Write the result in standard form.

Write the complex conjugate of

3 + 5i
and express the result in standard form
3 + 5i Conjugate is 3 - 5i

Write the complex conjugate.

Write the complex conjugate of

2 - 6i
and express the result in standard form
2 - 6i Conjugate is 2 + 6i

Write the complex conjugate.

Write the complex conjugate of

15i
and express the result in standard form
15i Conjugate is - 15i

Write the complex conjugate.

Perform the operation and

write the result in standard form

Multiply the numerator and
denominator by the conjugate of the
denominator of the original quotient.
Perform the multiplication of the two
fractions.
Perform the multiplication in the
numerator and perform the
multiplication in the denominator.

Simplify the denominator.

Write the result in standard form.

Perform the operation and

write the result in standard form

Multiply the numerator and
denominator by the conjugate of the
denominator of the original quotient.
Perform the multiplication of the two
fractions.
Perform the multiplication in the
numerator and perform the
multiplication in the denominator.
Simplify the numerator and
denominator.
Write the result in standard form.
Perform the operation and

write the result in standard form

(3 + 2i)(3 -2i)

Name the common denominator to be

used to combine the two fractions into
one fraction.
Write each fraction as an equivalent
fraction with this common denominator.
Combine the two fractions into one
fraction.

Simplify the numerator and the

denominator.
Write the result in standard form.
Solve the quadratic equation
3x2 + 9x + 7 = 0
using the quadratic formula.
Recall the quadratic formula
Substitute the values for a, b, and c into this
formula.
Simplify the expression under the radical

Rewrite this expression using complex number

notation.

Write the two answers in standard form.

Solve the quadratic equation

y2 - 2y + 2= 0
using the quadratic formula.
Recall the quadratic formula
Substitute the values for a, b, and c into this
formula.

Simplify the expression under the radical

Rewrite this expression using complex number

notation.
Write the two answers in standard form.