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Solving Quadratic Equations:
     Solving with the Quadratic Formula (page 4 of 6)

Sections: Solving by: factoring, taking roots, completing the square, using the formula, graphing

Somebody noticed that, when he solved by completing the square, he was always doing the exact same steps in the exact same order. So he made a formula out of it. This formula is the Quadratic Formula:

For ax² + bx + c = 0, the value of x is given by

The nice thing about the Quadratic Formula is that, unlike factoring, for instance, the Quadratic Formula always works. That is, there are some quadratics (most of them, actually) that do not factor, so you can't solve them by factoring. But the Quadratic Formula will always spit out an answer, whether the quadratic was factorable or not.

(I have a lesson on the Quadratic Formula, which gives examples and shows the connection between the discriminant (the stuff inside the square root), the number and type of solutions of the quadratic equation, and the graph of the related parabola. So I'll just do one example here. If you need further instruction, study the lesson at the above hyperlink.)

Let's try that last problem from the previous section again, but this time we'll use the Quadratic Formula:

• Use the Quadratic Formula to solve x² – 4x – 8 = 0.

Looking at the coefficients, I see that a = 1, b = –4, and c = –8. I'll plug them into the Formula, and simplify. I should get the same answer as before:

Then the solution is 

The nice thing about the Quadratic Formula (as compared to completing the square) is that you're just plugging into a formula. There are no steps to remember, and there are much fewer opportunities for mistakes. Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

Memorize the Quadratic Formula. (I don't care if your teacher says she's going to give it to you on the test; memorize it anyway, because you'll need it later. It's not that long, and there's even a song to help you remember it.)

When using the Formula, make sure you are careful not to omit the "±" sign, and be careful with the fraction line (don't draw it as being only under the square root; it's under the initial "–b", too). And, though many of your quadratics will start with "x²" so a = 1, don't forget that the denominator of the Formula is "2a", not just "2"; that is, when the leading term is something like "5x²", you will need to remember to put the "a = 5" value in the denominator. Take the time to be careful, because as long as you do your work neatly, the Quadratic Formula will give you the right answer every time.

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