Methods for Solving Quadratic Equations

Quadratic equations are of the form ax + bx + c = 0, where a 0

Quadratics may have two, one, or zero real solutions .

1. Completing the Square

If the quadratic equation is of the form ax + bx + c = 0, where a 0 and the quadratic expression is not factorable, try completing the square.

Example: x + 6x - 11 = 0

**Important: If a 1, divide all terms by “a” before proceeding to the next steps.

Move the constant to the right side	x + 6x = 11
Find half of b, which means
Find :	3 = 9
Add to both sides of the equation	x + 6x + 9 = 11 + 9
Factor the quadratic side	(x + 3)(x + 3) = 20
(which is a perfect square because you just made it that way!)
Then write in perfect square form	(x + 3)= 20
Take the square root of both sides
Solve for x	Simplify the radical
	This represents the exact answer. Decimal approximations can be found using a calculator.

2. Quadratic Formula

Any quadratic equation of the form ax + bx + c = 0, where a 0 can be solved for both real and imaginary solutions using the quadratic formula:

Example: x + 6x - 11 = 0 (a = 1, b = 6, c = -11)

Substitute values into the quadratic formula:

Simplify the radical

This is the final simplified EXACT answer.