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# Algebra and Probability Study Guide 2 for McGraw-Hill's ASVAB (page 3)

By McGraw-Hill Professional
Updated on Jun 26, 2011

A quadratic equation is one that is written in the form ax2 + bx + c, where a, b, and c are numbers.

To solve an equation in this form for x, set the expression equal to zero. Note that x will have more than one value.

• Put all the terms of the expression on one side of the = sign and set it equal to zero.
• Factor the equation.
• Set each factor equal to zero.
• Solve the equations.

### Algebraic Fractions

An algebraic fraction is a fraction containing one or more unknowns.

Reducing Algebraic Fractions to Lowest Terms

To reduce an algebraic fraction to lowest terms, factor the numerator and the denominator. Cancel out or divide common factors.

Example

Reduce to lowest terms:

Example

Reduce to lowest terms:

Adding or Subtracting Algebraic Fractions with a Common Denominator To add or subtract algebraic functions that have a common denominator, combine the numerators and keep the result over the denominator. Reduce to lowest terms.

Examples

Adding or Subtracting Algebraic Fractions with Different Denominators To add or subtract algebraic fractions that have different denominators, examine the denominators and find the least common denominator. Then change each fraction to the equivalent fraction with that least common denominator. Combine the numerators as shown in the previous section. Reduce the result to lowest terms.

Examples

Note that abis the least common denominator.

In this example, 12xis the least common denominator, as 4xand 6xboth divide into it.

Note that you have to multiply in order to make each term contain the least common denominator.

Multiplying Algebraic Fractions When multiplying algebraic fractions, factor any numerator and denominator polynomials. Divide out common terms where possible. Multiply the remaining terms in the numerator and denominator together. Be sure that the result is in lowest terms.

Examples

Multiply the following fractions:

Divide out common terms, then multiply. Reduce to lowest terms.

Multiply the following fractions:

Dividing Algebraic Fractions To divide algebraic fractions, follow the same process used to divide regular fractions: invert one fraction and multiply.

Examples

Divide the following fractions:

Divide the following fractions:

### Graphing on a Number Line

You can represent a number as a point on a number line, as shown in the following examples. Representing a number on a number line is called graphing. Note that whole numbers on the line are equally spaced. Note too that in these examples, both positive and negative numbers are represented.

On the number line, positive numbers are shown to the right of zero. Negative numbers are shown to the left of zero. The positive number +3 is three units to the right of zero. The negative number –2 is two units to the left of zero.

### Graphing on a Coordinate Plane

A coordinate plane is based on an xaxis (horizontal number line) and a yaxis (vertical number line). The axes intersect at their zero points. This point of intersection is called the origin. Every point on the plane has both an xcoordinate and a ycoordinate. The xcoordinate tells the number of units to the right of the origin (for positive numbers) or to the left of the origin (for negative numbers). The ycoordinate tells the number of units above the origin (for positive numbers) or below the origin (for negative numbers).

The coordinates of each point are often shown in what is called an ordered pairof numbers. An ordered pair looks like this: (2, 3). In every ordered pair, the first number is the xcoordinate, and the second number is the ycoordinate. So the ordered pair (2, 3) identifies a point with an xcoordinate of 2 and a ycoordinate of 3. The point is located at the intersection of the vertical line that is 2 units to the right of the origin (x = +2) and the horizontal line that is 3 units above the origin (y = +3). The point (2, –3) is located at the intersection of the vertical line that is 2 units to the right of the origin (x = +2) and the horizontal line that is 3 units below the origin (y = –3). The origin is identified by the ordered pair (0, 0).

The xand yaxes separate the graph into four parts called quadrants.

• Points in Quadrant I have positive numbers for both the xand the ycoordinates.
• Points in Quadrant II have a negative number for the xcoordinate but a positive number for the ycoordinate.
• Points in Quadrant III have negative numbers for both the xand the ycoordinates.
• Points in Quadrant IV have a positive number for the xcoordinate but a negative number for the ycoordinate.

Examples

The graph below shows the following points:

(2,3), (–3,2), (–4, –4), and (0,–2)

The graph below shows the following points:

A(4,–2), B(–1,1), C(3,3), and D(–4,–3).

Graphing Equations on the Coordinate Plane An equation with two variables xand ycan be graphed on a coordinate plane. Start by plugging in values for either xor y. Then solve the equation to find the value of the other variable. The xand y values make ordered pairs that you can plot on the graph.

Examples

Graph the equation x + y = 4

Solving for x, the equation becomes x = 4 – y

If y = 1, then x = 3.

If y = 2, then x = 2.

If y = 3, then x = 1.

If y = 4, then x = 0.

Plot the ordered pairs (3,1), (2,2), (1,3), and (0,4) on a graph. If you connect the points, you will see that the result is a straight line.

Graph the equation yx2 = 2

y = 2 + x2

If x = 0, then y = 2.

If x = 1, then y = 3.

If x = 2, then y = 6.

If x = 3, then y = 11.

If x = 4, then y = 18.

Graph these points and connect the points with a line. Note that when you connect the points, you get a curved line.

Probability

When every event in a set of possible events has an equal chance of occurring, probability is the chance that a particular event (or "outcome") will occur. Probability is represented by the formula

Let's say you have a spinner with an arrow that spins around a circle that is divided into six equal parts. The parts are labeled from 1 to 6. When you spin the arrow, what is the probability that it will land on the part labeled 4? Following the formula:

Let's take that same spinner. What is the probability that the arrow will land on the number 2 or the number 3 when it is spun? Using the formula:

Examples

Solve: The National Fruit Growers' Association is conducting a random survey asking people to tell their favorite fruit. The chart shows the results so far.

What is the probability that the next randomly selected person will say that pears are his or her favorite fruit?

To solve probability problems, follow the word problem solution procedure outlined in Chapter 9.

Procedure

What must you find?The probability that a certain event will occur

What are the units?Fractions, decimals, or percents

What do you know?The number of people selecting each fruit as their favorite.

Create an equation and solve

Substitute values and solve.

Number of positive outcomes = 125 people who named pears as their favorite fruit

Number of possible outcomes = all people surveyed = 236 + 389 + 250 + 125 = 1,000

Solve: A box is filled with 25 black balls, 50 white balls, and 75 red balls. If Wendell reaches into the box and picks a ball without looking, what is the probability that he will pick a black or a white ball?

Procedure

What must you find?The probability that either of two events will occur

What are the units?Fractions, decimals, or percents

What do you know?How many of each kind of ball are in the box

Create an equation and solve.

Substitute values and solve.

Number of positive outcomes = number of black balls + number of white balls = 25 + 50 = 75

Number of possible outcomes = 25 + 50 + 75 = 150

Practice questions for this study guide can be found at:

Algebra and Probability Practice Problems for McGraw-Hill's ASVAB