Algebra: The Quadratic Formula
The Quadratic Formula
The final method you can use to solve quadratic equations is called the quadratic formula, and it, like the completing-the-square method, will also solve any quadratic equation. Even better, it's very easy to use.
All you have to do is set the quadratic equation equal to 0, and it will look something like this:
- ax2 + bx + c = 0, where a, b, and c are real numbers
To get the solutions to the equation, just plug the coefficients a, b, and c into the quadratic formula:
I know that formula looks ugly at first, but you're going to have to memorize it. If you need help doing that, check out my web page www.calculus-help.com and click on the "Fun Stuff " section; I wrote a little song (parental warning: song contains comedic violence) to help you remember it.
Example 3: Solve the equation using the quadratic formula.
- 2x2 -5x = -1
Solution: To set the equation equal to 0, into the form ax2 + bx + c = 0, add 1 to both sides.
- 2x2 -5x + 1 = 0
So, a = 2, b = -5, and c = 1. Plug these values into the quadratic formula.
How'd You Do That?
Wondering where the quadratic formula comes from? Is it, too, the product of insect waste? Actually, in a manner of speaking, it is. The quadratic formula is the solution to the equation ax2 + bx + c = 0 when it's solved by completing the square.
The solutions to the quadratic equation 2x2 - 5x = -1 are
You could also rewrite the solutions as
but since none of those fractions can be reduced (simplified), there's no need to.
You've Got Problems
Problem 3: Solve the equation from Problem 2 (x2 + 6x - 3 = 0) again, this time using the quadratic formula; verify that you get the same answer as before.
Excerpted from The Complete Idiot's Guide to Algebra 2004 by W. Michael Kelley. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc.
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