he solutions to a quadratic equation can be found directly from the quadratic formula.
| The equation
ax2 + bx + has solutions
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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:
Given:  |
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Divide through by a: |
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| Move the constant term to the right side:  |
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| Add the square of one-half the coefficient of x to both sides:  |
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| Factor the left side (which is now a perfect square), and rearrange the right side:  |
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| Get the right side over a common denominator:  |
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| Take the square root of both sides (remembering to use plus-or-minus):  |
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Solve for x: |
