Review for Test 2. The Imaginary Unit The imaginary unit is defined as. In other words, has the property that its square is :. Square Roots of Negative.

Review for Test 2

The Imaginary UnitThe imaginary unit is defined as . In other words, has the property that its square is : .

Square Roots of Negative NumbersIf a is a positive real number, . Note that by this definition, and by a logical extension ofexponentiation, .

The Imaginary Unit and its Propertiesi

ii 1i

i2 1i

a i a

2 22i a i a a

1

Powers of i

The Powers of i follow a repeating pattern:i1 = ii2 = -1i3 = -ii4 = 1

The Algebra of Complex Numbers

When faced with a complex number, the goal is to write it in the form . a bi

Simplifying Complex Expressions

Step 1: Add, subtract, or multiply the complex numbers, as required, by treating every complex number as a polynomial expression. Remember, though, that is not actually a variable. Treating as a binomial in is just a handy device.

Step 2: Complete the simplification by using the fact that .

a bia bi

ii

2 1i

Simplify the following complex number expressions.

Example: The Algebra of Complex Numbers

2 1 2i i

4 3i

The product of two complex numbers leads to four products via the distributive property.

The Algebra of Complex Numbers

Division of Complex Numbers:In order to rewrite a quotient in the standard form , we make use of the following observation:

Given any complex number , the complex number is called its complex conjugate.

We simplify the quotient of two complex numbers by multiplying the numerator and denominator of the fraction by the complex conjugate of the denominator.

a bi 2 2 2 2 2a bi a bi a abi abi b i a b

a bia bi

Types of Equations

There are three types of equations:

1. A conditional equation has a countable number of solutions. For example, x + 7 = 12 has one solution, 5. The solution set is {5}.

2. An identity is true for all real numbers and has an infinite number of solutions. For example,

is true for all real number values of . The solution set is R.

3. A contradiction is never true and has no solution. For example, is false

for any value of . The solution set is Ø.

x6x x

x

2( 1)x x x x

Solving Linear Equations

To solve a linear equation (in x): 1. Simplify each side of the equation separately by removing any

grouping symbols and combining like terms. 2. Add or subtract the same expression(s) on both sides of the

equation in order to get the variable term(s) on one side and the constant term(s) on the other side of the equation and simplify.

3. Multiply or divide by the same nonzero quantity on both sides of the equation in order to get the numerical coefficient of the variable term to be one.

4. Check your answer by substitution in the original equation.

Solving Absolute Value Equations

The absolute value of any quantity is either the original quantity or its negative (opposite). This means that, in general, every occurrence of an absolute value term in an equation leads to two equations with the absolute value signs removed, if c > 0. Note: if c < 0, it has no solution.

ax b c means

ax b c or ax + b = -c

Example: Solving Absolute Value Equations

Step 1: Rewrite the absolute value equation without absolute values.

3 2 5x

3 2 5x

3 7x

7

3x 1x

Step 2: Solve the two equations

or 3x – 2 = -5

or 3x = -3

or

Solve:

Example: Solving Formulas for One Variable

h

22 2S r rh

22

2

S rh

r

22

2

S rh

r

22 2S r rh . Solve for .

Distance and Interest Problems

Good examples of linear equations arise from certain distance and simple interest problems.

The basic distance formula is where is distance traveled at rate for time .

The simple interest formula is where is the interest earned on principal invested at rate for time .

d rtd r t

I PrtIP r t

Example: Distance Problems

Two trucks leave a warehouse at the same time. One travels due west at an average speed of 61 miles per hour, and the other travels due east at an average speed of 53 miles per hour. After how many hours will the two trucks be 456 miles apart?

1 2 1 1 2 2d d rt r t

1 2 456d d 1 61r 2 53r

456 61 53t t

Combine like terms, and solve.456 114t

4t hours

Given.

Plug values in.

Solving Linear Inequalities

Cancellation Properties for InequalitiesThroughout this table, and , represent algebraic expressions. These properties are true for all inequalities.

Property Description

A B A C CB

,A B C

Adding the same quantity to both sides of an inequality results in an equivalent inequality.

If both sides of an inequality are multiplied by a positive quantity, the sense of the inequality is unchanged.

If both sides of an inequality are multiplied by a negative quantity, the sense of the inequality is reversed.

0, C CA B B CA If

0, C CA B B CA If

Example: Linear Inequalities

Solve the following linear inequality.

10 4( 2) (2 )x x Step 1: Distribute. 10 4 8 2x x

4 18 2x x

5 20x Step 3: Divide by . Note the reversal of the inequality sign.

5

4x

Solution is 4,

Step 2: Combine like terms.

Solving Compound Linear Inequalities

A compound inequality is a statement containing two inequality symbols, and can be interpreted as two distinct inequalities joined by the word “and”.

For example, in a course where the grade depends solely on the grades of 5 exams, the following compound inequality could be used to determine the final exam grade needed to score a B in the course.

92 65 71 8080 90

5

x

Example: Solving Compound Inequalities

Solve the compound inequality from the previous slide. 92 65 71 80

80 905

x

Step 1: Multiply all sides by . 5

400 308 450x

Step 2: Subtract from all sides.

92 142x

Solution is 92,142

Note: If this compound inequality relates to test scores, as indicated on the previous slide, the solution set is , assuming is the highest score possible.

92,100

308

100

Solving Absolute Value Inequalities

An absolute value inequality is an inequality in which some variable expression appears inside absolute value symbols. can be interpreted as the distance between and zero on the real number line. This means that absolute value inequalities can be written without absolute values as follows, assuming is a positive real number:

x

x a a x a

x a x a x aorand

x

a

Example: Solving Absolute Value Inequalities

Solve the following absolute value inequality.5 3 2 9y

5 3 7y Step 1: Subtract . 2

7 5 3 7y Step 2: Rewrite the inequality without absolute values.

4 5 10y Step 3: Solve as compound inequality. 4

25

y

4

Solution is ,25

Example: Solving Absolute Value Inequalities

Solve the following absolute value inequality and graph the solution.

6 2 8x

6 2 8x 6 2 8x 2 14x 2 2x

7x 1x

Solution is , 1 7,

1

7

oror

or

Solving Quadratic Equations by Factoring

• The key to using factoring to solve a quadratic equation is to rewrite the equation so that appears by itself on one side of the equation.

• If the trinomial can be factored, it can be written as a product of two linear factors and .

• The Zero-Factor Property then implies that the only way for to be is if one (or both) of and is .

0

2ax bx c A B

2ax bx c 0 AB 0

Example: Solving Quadratic Equations by Factoring

Solve the quadratic equation by factoring.

2 3 2

5 5

xx

Step 1: Multiply both sides by .5

Step 2: Subtract from both sides so is on one side.

25 3 2x x 25 3 2 0x x 2

0

Step 3: Factor and solve the two linear equations.

(5 2)( 1) 0x x 5 2 0x 1 0x

2

5x

1x

or

or

Solving “Perfect Square” Quadratic Equations

In some cases where the factoring method is unsuitable, the solution to a second-degree polynomial can be obtained by using our knowledge of square roots. If is an algebraic expression and if is a constant:

If a given quadratic equation can be written in the form we can use the above observation to obtain two linear equations that can be easily solved.

A c

2A c A cimplies

2A c

Example: “Perfect Square” Quadratic Equations

Solve the quadratic equation by taking square roots.

27 25 0x

27 25x

7 25x

7 5x i

7 5x i

In this example, taking square roots leads to two complex number solutions.

The Quadratic Formula

The solutions of the equation are:

Note: • The equation has 2 real solutions if .• The equation has 1 real solution if .• The equation has 2 complex solutions (which are

conjugates of one another) if .

2 0ax bx c

2 4

2

b b acx

a

2 4 0b ac 2 4 0b ac

2 4 0b ac

Example: The Quadratic Formula

Solve using the quadratic formula.

24 7 15x x

2 15 074x x a b c

24

2

7 7 4 1

4

5x

2 4

2x

cb b a

a

7 289

8x

7 17

8x

53,

4x

Solving Quadratic-Like Equations

An equation is quadratic-like, or quadratic in form, if it can be written in the form

Where , , and are constants, , and is an algebraic expression. Such equations can be solved by first solving for and then solving for the variable in the expression . This is the Substitution Method.

2 0aA bA c a b c 0a A

AA

Example: Solving Quadratic-Like Equations

Solve the quadratic-like equation. 22 23 2 3 8 0 x x x xStep 1: Let

and factor.

2 3A x x 2 2 8 0A A

4 2 0 A A

4A 2A2 3 4x x 2 3 2x x

2 3 4 0x x 2 3 2 0x x

4 1 0x x 2 1 0x x

4 or 1x x 2 or 1x x

Step 2: Replace with

and solve for .

A2 3x x

x

or

or

or

or

or

Example: General Polynomial Equations

Solve the equation by factoring. 4 4x 4 4 0x Step 1: Isolate

on one side and factor.

2 22 2 0x x

Step 2: Set both equations equal to and solve.

2 2x 2 2x

2 2 0x 2 2 0x

2x

0

0

ororor2x i

Operations with Rational Expressions

• To add or subtract two rational expressions, a common denominator must first be found.

• To multiply two rational expressions, the two numerators are multiplied and the two denominators are multiplied.

• To divide one rational expression by another, the first is multiplied by the reciprocal of the second.

• No matter which operation is being considered, it is generally best to factor all the numerators and denominators before combining rational expressions.

Example: Add/Subtract Rational Expressions

Subtract the rational expression.

2 2

5 2 2

5 6 4

x x

x x x

5 2 2

2 3 2 2

x x

x x x x

2 35 2 2

2 2 3 3 2 2

x xx x

x x x x x x

2 25 8 4 2 6

2 2 3 2 2 3

x x x x

x x x x x x

23 14 4

, 2, 2, 32 2 3

x xx

x x x

Step 1: Factor both denominators.

Step 2: Multiply to obtain the least common denominator (LCD) and simplify.

Example: Multiply Rational Expressions

Multiply the rational expression. 2

2

2 3 2

2 3 4

x x x

x x x

3 1 2

2 1 4

x x x

x x x

3 1 2

2 1 4

x x x

x x x

3 2, 4, 2,1

2 4

x xx

x x

Solving Rational Equations

• A rational equation is an equation that contains at least one rational expression, while any non-rational expressions are polynomials.

• To solve these, we multiply each term in the equation by the LCD of all the rational expressions. This converts rational expressions into polynomials, which we already know how to solve.

• However, values for which rational expressions in a rational equation are not defined must be excluded from the solution set.

Work-Rate Problems

There are two keys to solving a work-rate problem:1. The rate of work is the reciprocal of the time needed

to complete the task. If a given job can be done by a worker in units of time, the worker works at a rate of jobs per unit of time.

2. Rates of work are “additive”. This means that two workers working together on the same task have a combined rate of work that is the sum of their individual rates.

x1

x

1

Time

1

Ti

1

Ti me 1 Tome ge

er2 th

Rate 1 Rate 2 Rate Together

Example: Work-Rate Problem

One hose can fill a swimming pool in 10 hours. The owner buys a second hose that can fill the pool in half the time of the first one. If both hoses are used together, how long does it take to fill the pool?The work rate of the first hose is

1

10

The work rate of the second hose is 1

5 1 1 1

10 5 x Step 1: Set up the problem.

Step 2: Multiply both sides by the LCD , and solve. 10x

2 10x x

3 10x 1

33

x hours