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

Somebody (possibly in seventh-century India) was solving a lot of quadratic equations by completing the square. At some point, he noticed that he was always doing the exact same steps in the exact same order for every equation. Taking advantage of the one of the great powers and benefits of algebra (namely, the ability to deal with abstractions, rather than having to muck about with the numbers every single time), he made a formula out of what he'd been doing:

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

The nice thing about the Quadratic Formula is that the Quadratic Formula always works. There are some quadratics (most of them, actually) that you can't solve 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 fewer opportunities for mistakes. Copyright © Elizabeth Stapel 2002-2011 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 next test! Memorize it anyway, because (warning!) you'll need it later. It's not that long, and there's even a song to help you remember it.

Advisory: 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" part, 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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