The solutions to a quadratic equation can be found directly from the quadratic formula.
The equation ax^{2} + 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 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 |
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 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: |
_{} |