Difference between revisions of "Completing the square"
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| − | + | '''Completing the square''' is a method for solving for the roots of the general [[quadratic equation]]: | |
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| + | <math>ax^2 + bx + c = 0</math>, where <math>a \ne 0</math> | ||
| + | |||
| + | It is first taught using equations with "friendly" numbers in place of ''a'', ''b'', and ''c'' to get the student used to the process. | ||
| + | |||
| + | :What one does is add and multiply by various carefully chosen constants to create an equation of the form: | ||
| + | |||
| + | :<math>d^2x^2 + 2dex + e^2 = f</math> where ''d'', ''e'' and ''f'' are constants expressed in ''a'', ''b'', and ''c''. | ||
| + | |||
| + | :This resolves to: | ||
| + | |||
| + | ::<math>(dx+e)^2 = f</math> (grouping) | ||
| + | ::<math>dx + e = \pm \sqrt{f}</math> (take square root) | ||
| + | ::<math>dx = -e \pm \sqrt{f}</math> (subtract e) | ||
| + | ::<math>x = \frac{-e \pm \sqrt{f}}{d}</math> (divide by e) | ||
| + | |||
| + | By then applying the process to the general equation, we can derive the [[quadratic formula]]: | ||
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| + | :<math>ax^2 + bx + c = 0</math> (given) | ||
| + | |||
| + | :<math>4a^2x^2 + 4abx + 4ac = 0</math> (multiply by 4a) | ||
| + | |||
| + | :<math>4a^2x^2 + 4abx = -4ac</math> (subtract 4ac) | ||
| + | |||
| + | :<math>4a^2x^2 + 4abx + b^2 = -4ac + b^2</math> (add b^2) | ||
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| + | :<math>(2ax + b)^2 = b^2 - 4ac</math> (group each side) | ||
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| + | :<math>2ax + b = \pm \sqrt{b^2 - 4ac}</math> (take square root, allow for both roots) | ||
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| + | :<math>2ax = -b \pm \sqrt{b^2 - 4ac}</math> (subtract b) | ||
| + | |||
| + | :<math>x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}</math> (divide by 2a) | ||
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| + | We can now determine the real or imaginary roots of any quadratic equation by simply inserting ''a'', ''b'', and ''c'' into the formula. | ||
| + | [[category:mathematics]] | ||
| + | [[category:algebra]] | ||
Revision as of 05:16, August 30, 2011
Completing the square is a method for solving for the roots of the general quadratic equation:
, where 
It is first taught using equations with "friendly" numbers in place of a, b, and c to get the student used to the process.
- What one does is add and multiply by various carefully chosen constants to create an equation of the form:
where d, e and f are constants expressed in a, b, and c.
- This resolves to:
(grouping)
(take square root)
(subtract e)
(divide by e)
By then applying the process to the general equation, we can derive the quadratic formula:
- (given)
(multiply by 4a)
(subtract 4ac)
(add b^2)
(group each side)
(take square root, allow for both roots)
(subtract b)
(divide by 2a)
We can now determine the real or imaginary roots of any quadratic equation by simply inserting a, b, and c into the formula.
