Regents Examination in Algebra I: A Brief Review of Algebra Skills

Key Ideas
Certain algebra skills show up frequently in our study of geometry. They are tools used to complete the evaluation of geometric relationships. Procedures for operations with radicals, solving linear equations, multiplying polynomials, solving quadratic equations, and solving proportions as well as word problem strategies are briefly reviewed.

Operations with Radicals
The square root of a number is the number that when multiplied by itself results in the original number. It is represented with the square root, or radical, symbol . The number under the radical symbol is the radicand. An example of a radical expression is . The 3 is the coefficient, and the 5 is the radicand.
 

Adding and Subtracting Radicals:

Add or subtract the coefficient if the radicands are the same, otherwise the radicals cannot be combined. For example, only the

terms can be combined in the following equation.

 

Multiplying and Dividing Radicals: Multiply or divide the coefficients and then multiply or divide the radicands.

 

Simplifying Radicals—A radical is said to be in simplest form when the following 3 conditions are met.

1. No perfect square factors appear in the radicand.
2. No fractions appear in the radicand.
3. No radicals appear in the denominator.

Remove perfect square factors from the radicand by factoring the radicand using the largest perfect square factor. Then take the square root of the perfect square factor. In the radical expression below, 12 is factored into 4 · 3.

Then

is simplified to 2.

Fractions in the radicand can be rewritten as the quotient of two radicals as shown below.

Remove radicals in the denominator by multiplying the numerator and denominator by the radical in the denominator.

This is called “rationalizing the denominator.”

Math Fact
Taking the square root and squaring are inverse operations.

Taking the square root of a number and then squaring it results in the original number, as in
.


Example 1
Express

in simplest radical form
Solution:


Example 2
Express

in simplest radical form
Solution:



Math Fact
Expressing radicals in simplest radical form make it easier to compare radical expressions. In geometry, we often want to determine if two measures are equal or satisfy a particular inequality. Once all the radicals have been completely simplified, comparing radicals is just a matter of comparing the coefficients. Radicals will frequently show up when working with solving quadratic equations, when using the Pythagorean theorem, or with the distance formula.


Solving Linear Equations
Linear equations are equations that involve the variable raised to the first power only. They can be solved using the following steps.

Apply the distributive property to terms with parentheses.
Eliminate fractions by multiplying both sides by the denominator of any fraction, or the greatest common denominator if there are several fractions.
Combine like terms on each side of the equal sign.
Isolate the variable by undoing additions/subtractions and then multiplications.


Example 1
Solve 8x + 6 = 2x + 4(x + 5)
Solution:
 

Example 2
Solve 

Solution:

Multiplying Polynomials
When multiplying monomials, multiply the coefficients and multiply the variables. When multiplying powers of the same variable, use the rule “keep the base and add the powers.”

Example 1
Multiply

Solution:

When multiplying binomials, use the double distributive property by applying the vertical method, box method, FOIL (first-outer-inner-last), or any other technique you may have learned.

Example 2
Multiply (4x + 2) (5x + 6)
Solution: Use FOIL.


Example 3
Multiply (3x − 7)(x + 2)
Solution: Use the vertical method.



Factoring and Solving Quadratic Equations
Quadratic equations have second-order, or , terms as the highest power of x. Solving quadratic equations requires factoring. The procedure is as follows:

Get all terms on one side.
Factor.
Apply the zero product rule.
Solve for x.


Math Fact
The zero product rule states that if a product of factors equals zero, then each factor may be individually set equal to zero and solved to find a solution to the equation.


Some Factoring Methods
Greatest Common Factor: If a common factor exists among all terms, divide all terms by that factor. Put the new terms inside parentheses, and move the divided factor outside the parentheses. If a factor is still not linear, use another method on that factor.

Example 1
Solve

Solution:

 

Grouping with a = 1: For the quadratic equation
, find numbers

and

such that

and
. The equation factors to
. From here, apply the zero product theorem.

Example 2
Solve

Solution:

 

Difference of Perfect Squares: Quadratics in the form

factor into

Once factored, set each factor equal to zero and solve for x.

Example 3
Solve

Solution:

 

Completing the Square: Completing the square can be used on any trinomial with the form
.
Rewrite the equation as

Add the quantity

to both sides of the equation.
Factor the left side, which will be a perfect square.
Take the square root of both sides, and solve for x.

Remember, there will be a positive and negative root when taking the square root. So there will be two solutions.

Example 4
Solve

Solution:

 

Quadratic Formula: The solution to any quadratic equation of the form

can be found using the quadratic formula.


Example 5
Solve

Solution:





Math Fact
Some geometric relationships result in a quadratic equation that must be solved in order to find the measure of an angle or segment. The quadratic will give two solutions, and both must be checked for consistency with the problem. Lengths or angle measure in this course will always be positive. If either solution results in a negative length or angle, that solution is thrown out. If both solutions lead to an acceptable answer, the problem has two solutions.

Two solutions often correspond to a situation where two different geometric configurations could lead to the relationship modeled in the equation.


Solving Proportions
A proportion is an equation involving two ratios. They can be solved using the fact that the cross products must be equal.


Example 1
Solve

Solution:


Example 2
Solve

 

Solution:

Word Problem Strategies
Word problems in geometry may involve phrases that describe a relationship between two figures or measures. Some common phrases and their algebraic translations are shown below.

Phrase	Algebra
x is two more than y x is two greater than y	x = y + 2
x is two less than y	x = y − 2
x is twice y x is double y	x = 2y
x is half y
Three quantities are in a 1 : 2 : 3 ratio	Represent the quantities as x, 2x, and 3x

The following are some good general strategies for solving word problems:

Make a sketch and label it.
Underline or highlight key words and definitions, such as bisector, midpoint, and so on.
Underline phrases to be translated into mathematical expressions.
Identify what the question is asking—the value of a variable, the measure of an angle or segment, an explanation or justification, and so on.


Example 1
Write an expression that represents “12 less than double a number.”
Solution: Let the number equal
.


Example 2
Three integers are in a 4 : 7 : 9 ratio. If their sum equals 60, what are the numbers?
Solution: Let the integers equal 4x, 7x, and 9x.