# Quadratic Equations

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

## Rule

To solve an equation such as “x$^2$ + 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.
x$^2$ - 3x = -3
                Start with the initial equation.
x$^2$ -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

x$^2$ - y$^2$ factors out to (x - y)(x + y). This is known as the rule of “difference of two squares.”
Thus, x$^2$ - 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. x$^2$ + 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 x$^2$ + 6x + 9 = 0 and multiply up to the third number. Observe more examples:
Original equation Set it equal to zero Factor Solution
x$^2$ + 4 = -4x x$^2$ + 4x + 4 = 0 (x + 2)(x + 2) x = -2
x$^2$ -4x = -3 x$^2$ - 4x + 3 = 0 (x - 3)(x - 1) x = 3, 1
x$^2$ - 1 = x x$^2$ - 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 x$^2$ + 5 = -6x, then what is the value of x?
2. x$^2$ - 2x = 8. Solve for x.
3. If x$^2$ + 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.