# GMAT Problems – Quadratic Equations

**Quadratic equations** are among the most advanced algebra concepts tested on the GMAT. You will encounter quadratic equations on both Problem Solving and Data Sufficiency question types on the GMAT quantitative section. Fear not — they need not be scary! There are just a few basic things you must know.

### GMAT Quadratic Equations: The Basics

A quadratic equation is simply a polynomial equation of the second degree. Simply put, it’s an equation that contains a sqaure term. Quadratic equations follow this general form:

**ax² + bx + c = 0**

where ** x** represents a variable (or unknown), and

**a**,

**b**, and

**c**are constants where a ≠ 0.

Often on the GMAT you will be asked to find the * roots* of the equation, which is where x = 0. A quadratic equation will either have 2 roots, 1 root, or 0 roots. Check out this simple trick for determining how many real roots a quadratic equation has, which is particularly useful on data sufficiency questions involving quadratic equations.

Generally, the best way to find the roots of a quadratic equation is to factor it. Consider this sample quadratic equation:

**x² – x – 12 = 0**

What are the factors of this equation?

**Answer: (x+3)(x-4) = 0** because if you multiply those factors together using the FOIL method, it produces the original quadratic equation. If either factor = 0, then the entire equation will equal zero. Thus, to find the roots, set each factor equal to zero and solve for x:

If x + 3 = 0, then x = -3.

If x – 4 = 0, then x = 4.

So, **the roots of the equation are -3 and 4**. In other words, there are two (2) possible values for which the original quadratic equation will equal zero.

### On the GMAT

Rather than always trying to factor quadratic equations on the GMAT, you should recognize that the GMAT test makers like to test the following three (3) most **common quadratic equations:**

**x² – y² = (x-y)(x+y)****x² – 2xy + y² = (x-y)²****x² + 2xy + y² = (x+y)²**

If you see GMAT sample questions that fall into one of these common quadratic equation patterns, then you should be able to immediately recognize what the factors of the equation are, which will enable you to solve the question much more effectively and efficiently. Try applying that strategy to this sample GMAT question:

**Q: If x² – y² = 100, and if x + y = 2, then x – y =**

**(A) -2**

** (B) 10**

** (C) 20**

** (D) 50**

** (E) 100**

For a step-by-step explanation on how to solve this question, check out this short video:

Here is a full transcript of this video, for your convenience: