A Guide to the GRE/Quadratic Equations

= Quadratic Equations =

A quadratic equation is an equation such as x2 - 3x = 2, where x is multiplied by both itself and a constant.

Rule
'To solve an equation such as “x2 + 2x'' = -1”, adjust the equation so that a zero is on one side, then break the equation into factors such as (x + 1)(x + 1) = 0. The correct answer is the opposite of the numbers in the parentheses.'''

x2 - 3x = -3
 * Start with the initial equation.

x2 -3x + 3 = 0
 * Add 3 to both sides to set the equation equal to zero.

(x + ?)(x + ?) = 0
 * Break the equation into two “roots” or “factors.”

(x - 3)(x - 1) = 0
 * Find the two numbers which add up to the second number (the -3) and multiply to the third number (the 3).  These 	 		will be the numbers in the parentheses, that multiply to x in the original equation.

x = 3, 1

Most quadratic equations have two answers because they have different numbers in each factor. Some, however, like (x + 1)(x + 1), only have one answer.

Difference of Two Squares
x2 - y2 factors out to (x - y)(x + y). This is known as the rule of “difference of two squares.”

Thus, x2 - 9 would thus factor out to (x - 3)(x + 3). This is a special rule that is often tested on the GRE.

Factoring
“Factoring” is one of the most difficult parts of solving a quadratic equation. x2 + 6x + 9 = 0 will factor into (x + 3)(x + 3), because if multiplied out, (x + 3)(x + 3) equals the prior equation. The numbers in the parentheses will add up to second number in the equation x2 + 6x + 9 = 0 and multiply up to the third number. Observe more examples:

Original equation	Set it equal to zero	 Factor		Solution

x2 + 4 = -4x		x2 + 4x + 4 = 0		(x + 2)(x + 2)		x = -2

x2 -4x = -3		x2 - 4x + 3 = 0		(x - 3)(x - 1)		x = 3, 1

x2 - 1 = x		x2 - x - 2 = 0		(x - 2)(x + 1)		x = 2, -1 (find two numbers						that add to the middle						number and multiply to						the third number)

Practice
1. If x2 + 5 = -6x, then what is the value of x?

2. x2 - 2x = 8. Solve for x.

3. If x2 + 4x + 2 = -1, then what does x equal?

Answers to Practice Questions
1. -5, -1

x2 + 5 = -6x		Take the initial equation.

x2 + 6x + 5 = 0		Add 6x to both sides to set the equation equal to zero. Make sure that the equation is in the form x2 + bx + c = 0.

(x + ?)(x + ?)		Break the equation into factors. What two numbers add up to 			the middle number (6) and multiply to the last number (5)?

(x + 5)(x + 1) = 0	Plug the correct numbers into the equation.

x = -5, -1		x is the opposite of these numbers.

2. 4, -2

x2 - 2x = 8		Take the initial equation.

x -2x - 8 = 0		Subtract 8 from both sides to set the equation equal to zero. Make sure that the equation is in the form x2 + bx + c = 0.

(x + ?)(x + ?)		Break the equation into factors. What two numbers add 				up to the middle number (-2) and multiply to the last 				number (-8)?

(x - 4)(x + 2) = 0	Plug the correct numbers into the equation.

x = 4, -2		x is the opposite of these numbers.

3. -3, -1

x2 + 4x + 2 = -1	Take the initial equation.

x2 + 4x + 3 = 0	Add 1 to both sides to set the equation equal to zero. Make sure that the equation is in the form x2 + bx + c = 0.

(x + ?)(x + ?)		Break the equation into factors. What two numbers add up to 			the middle number (4) and multiply to the last number (3)?

(x + 3)(x + 1) = 0	Plug the correct numbers into the equation.

x = -3, -1		x is the opposite of these numbers.

Some quadratic equations cannot be factored and must be solved using a special formula, the “quadratic formula.” This formula is not tested on the GRE.