A quadratic equation is a type of mathematical equation where the highest power of x is two. One example of such an equation is: 4x + 5x + 3 = x - 5. It is difficult to solve such an equation since the methods that work on a x term do not work on an x term, and vice versa. One must either factor the quadratic (to break apart the x) or use the quadratic formula.
Using Factoring
Put all the terms on one side, preferably the side where the x term is positive. 
Factor the expression. For more information on how to do so, please consult the article How to Factor Second Degree Polynomials (Quadratic Equations). 
Set each factor equal to zero, as separate equations. 
Solve each equation independently. Improper fractions should not be written as mixed numbers, although it wouldn't be mathematically wrong to do so.
Using the Quadratic Formula
Put all the terms on one side of the equals sign, preferably the side where the x term will be positive. 
Identify the values of a, b, and c.a is the coefficient of x term, b is the coefficient of the x term, and c is the constant (doesn't have an x). Be sure to include the sign of the coefficient. 
Calculate the value of 4 times a times c. Why you do this will become clear on the next step. 
Write down the quadratic formula, which is: 
Substitute the values of a, b, c, and 4ac into the formula: 
Tidy up signs in the numerator, finish multiplying the denominator, and calculate b. Notice that even when b is negative, b is positive. 
Finish the part under the square root sign. This portion of the formula is called the "discriminant". Sometimes it is helpful to calculate beforehand since it predicts what kind of answer you will get from the formula. 
Simplify the square root. If the number under the radical symbol is a perfect square, you will get a whole number. If the number is not a perfect square, then simplify to its simplest radical version. If the number is negative, and you're sure it's supposed to be negative, then there is no answer to the problem. 
Split up the plus-or-minus into a plus option and a minus option. (This step applies only if the square root symbol has been simplified away.) 
Calculate the plus option and the minus option separately... 
...and take each one to lowest terms. Improper fractions do not need to be put as a mixed number, but you can.
Completing the Square
A different variation of the quadratic equation, this method may be easier to apply.
Ex: 2x - 12x - 9 = 0
- Put all the terms on one side, preferably the side where the a or x term is positive. 2x - 9 = 12x 2x - 12x - 9 = 0
- Move the c term or constant to the other side. 2x - 12x = 9
- Divide both sides by the coefficient of the a or x term if needed. x - 6x = 9/2
- Divide b by two and square this value. Add to both sides. -6/2 = -3 (-3) = 9 x - 6x + 9 = 9/2 + 9
- Simplify both sides. Factor one side (left in example). The factored form will be (x - b/2). Add like terms in the other (right in example). (x - 3)(x - 3) = 9/2 + 18/2 (x - 3) = 27/2
- Take the square root of both sides. Don't forget to add the plus or minus sign (±) to the constant side. x - 3 = ±v(27/2)
- Simplify the radical and solve for x. x - 3 = ±3v(6) ------- 2 x = 3 ±3v(6) ------- 2
Tips
If the number under the square root is not a perfect square, then the last few steps run a little differently. Here is an example: - As you can see, the radical sign did not disappear completely. Therefore, the terms in the numerator cannot be combined (because they are not like terms). There is no purpose, then, to splitting up the plus-or-minus. Instead, we divide out any common factors --- but ONLY if the factor is common to both of the constants AND the radical's coefficient.
