# Basic Algebra

Most everything in this section was tested in the GMAT, so my review will be brief. If your algebra skills are week, I strongly suggest that you find an old textbook and work practice problems. As a former calculus tutor, I can say that most “calculus problems” actually stem from poor algebra skills.

## Equations and Factoring

An *equation* is a statement that two expressions are equal. *Solving* the equation is the process of finding all values of the variables such that the equation is true.

For example, the *linear* equation, **3x + 4 = 25**, has exactly one solution which can be found with simple algebra:

3x + 4 = 25 - 4 -4 ___________ 3x = 21 __ __ 3 3 ___________ x = 7

Linear equations are ubiquitous in economics, operations management, and other business disciplines. It is no surprise that they have their own section in this review.

### Quadratic Equations

Probably the next most commonly encountered type of equation, after linear equations, is the *quadratic equation*. These equations have the general form **ax ^{2} + bx + c = 0**. Because a quadratic is a second order equation, it has two solutions. Note, however, that these solutions are not necessarily real numbers. There are several techniques for solving quadratics. In the “real world” you would probably solve them by graphing them or sticking them in a spreadsheet and using the solver. For this test, however, you need to know how to solve quadratics by

*factoring*and by using the

*quadratic formula.*

#### Solving Quadratic Equations by Factoring

*Factoring* is the process of breaking an expression into simpler terms that multiply together to produce the original expression.

Consider the quadratic equation **x ^{2} + x – 12 = 0**, which can be solved by factoring the expression on the left of the equals sign into two simpler terms:

**x ^{2} + x – 12 = (x + 4)(x – 3) = 0**

The solutions are therefore x = -4, and x = 3, because either (x + 4) or (x – 3) must evaluate to 0 in order for the equation to be true.

The acronym **FOIL** is a useful mnemonic for remembering how to factor these equations. It stand for First, Outer, Inner, Last.

If **Ax ^{2} + Bx + C = (__x + __)(__x + __)**, where the blank spaces represent coefficients that need to be found, then:

A = the product of the **F**irst two coefficients

B = the sum of the products of the **O**uter two coefficients and the **I**nner two coefficients

C = the product of the **L**ast two coefficients

Sometimes the equation is simple enough that we can figure out the FOIL coefficients in our head. Other times, a more formal approach is warranted. This approach is called the “AC method” or “grouping method in Algebra texts.

**AC / Grouping Method**

Given a trinomial of the form **Ax ^{2} + Bx + C**,

- Multiply A • C
- Find factors
*p*and*q*such that*pq = AC* - Split the middle term of the trinomial as a sum of
*p*and*q* - Factor by grouping

As an example, consider the equation **6x ^{2} + 5x – 4 = 0**. Multiplying the A and C terms gives us

**A • C = 6 • -4 = -24.**

Next, we make a table of all of the factors of -24, and their sums:

Factors |
Sum |
---|---|

1, -24 |
-23 |

-1, 24 |
23 |

2, -12 |
-10 |

-2, 12 |
10 |

3, -8 |
-5 |

-3, 8 |
5 |

4, -6 |
-2 |

-4, 6 |
2 |

The factors -3 and 8 add up to 5, which is our B coefficient.

Splitting the middle term, we have:

**6x ^{2} + 5x – 4 = 6x^{2} -3x + 8x – 4**

We can then factor this by grouping and solve for x:

**6x ^{2} -3x + 8x – 4 = 0**

**3x(2x – 1) + 4(2x – 1) = 0**

**(3x + 4)(2x – 1) = 0**

**x = {-4/3, 1/2}**

I will end my discussion of factoring by reminding you of three formulas which can be derived from the FOIL method and often come in handy for general algebra:

**(A + B) ^{2} = A^{2} + 2AB + B^{2}**

**(A – B) ^{2} = A^{2} – 2AB + B^{2}**

**(A + B)(A – B) = A ^{2} – B^{2}**

#### Solving Quadratic Equations with the Quadratic Formula

Many quadratic equations are simply too messy to solve by factoring. For instance, they might include decimals and/or rational numbers, or the solutions might be complex numbers. Luckily, we have a “brute force” method that always works: the quadratic formula.

The quadratic formula is definitely worth memorizing. To use it, just plug in the coefficients and solve for x, as in the following example:

## Inequalities

Inequalities are like equations but a <, >, <, or > replacing the = sign. Inequalities are solved just like equations, with one important difference. Whenever you multiply or divide an inequality by a negative number, reverse the direction of the inequality.

In textbooks and quizzes, inequalities often contain an absolute value term, which is surrounded by vertical bars (||). Should you encounter one of these problems, your best bet is to break it into two inequalities–one for if the term is positive and one for if it is negative–and solve separately. Hopefully, the following example will clarify:

Since this is an inequality in one variable, we can graph it on a number line:

## Sources

Beecher, J., Penna, J., & Bittenger, M. (2008). *College Algebra* [3rd Ed.]. Boston, MA: Pearson.

Blitzer, R. (2006). *Introductory Algebra* [4th Ed.]. Upper Saddle River, NJ: Pearson.