The Quadratic Formula

The solutions to a quadratic equation can be found directly from the quadratic formula.

The equation

ax2 + bx + c = 0

has solutions

The advantage of using the formula is that it always works. The disadvantage is that it can be more time-consuming 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 plus-or-minus 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 b2 – 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 b2 – 4ac > 0

There are two distinct real roots

If b2 – 4ac = 0

There is one real root

If b2 – 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:


Divide through by a:

Move the constant term to the right side:

Add the square of one-half 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 plus-or-minus):

Solve for x: