The Quadratic Formula
he solutions to a quadratic equation can be found directly from the quadratic formula.
ax2 + bx +
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 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
|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
|Take the square root of both sides
(remembering to use plus-or-minus):
|Solve for x:|