The Quadratic Formula
he solutions to a quadratic equation can be found directly from the quadratic formula.
The equation
ax^{2} + bx + has solutions 
The advantage of using the formula is that it always works. The disadvantage is that it can be more timeconsuming than some of the methods previously discussed. As a general rule you should look at a quadratic and see if it can be solved by taking square roots; if not, then if it can be easily factored; and finally use the quadratic formula if there is no easier way.
 Notice the plusorminus symbol (±) in the formula. This is how you get the two different solutions—one using the plus sign, and one with the minus.
 Make sure the equation is written in standard form before reading off a, b, and c.
 Most importantly, make sure the quadratic expression is equal to zero.
The Discriminant
The formula requires you to take the square root of the expression b^{2 }– 4ac, which is called the discriminant because it determines the nature of the solutions. For example, you can’t take the square root of a negative number, so if the discriminant is negative then there are no solutions.
If b^{2} – 4ac > 0  There are two distinct real roots 
If b^{2} – 4ac = 0  There is one real root 
If b^{2} – 4ac < 0  There are no real roots 
Deriving the Quadratic Formula
The quadratic formula can be derived by using the technique of completing the square on the general quadratic formula:
Given:  
Divide through by a:  
Move the constant term to the right side: 

Add the square of onehalf the coefficient of x to both sides: 

Factor the left side (which is now a perfect square), and rearrange the right side: 

Get the right side over a common denominator: 

Take the square root of both sides (remembering to use plusorminus): 

Solve for x: 