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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>
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| − | It is first taught using equations with "friendly" numbers in place of ''a'', ''b'', and ''c'' to get the student used to the process.
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| − | :What one does is add and multiply by various carefully chosen constants to create an equation of the form:
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| − | :<math>d^2x^2 + 2dex + e^2 = f</math> where ''d'', ''e'' and ''f'' are constants expressed in ''a'', ''b'', and ''c''.
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| − | :This resolves to:
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| − | ::<math>(dx+e)^2 = f</math> (grouping)
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| − | ::<math>dx + e = \pm \sqrt{f}</math> (take square root)
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| − | ::<math>dx = -e \pm \sqrt{f}</math> (subtract e)
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| − | ::<math>x = \frac{-e \pm \sqrt{f}}{d}</math> (divide by e)
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| − | 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)
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| − | :<math>4a^2x^2 + 4abx + 4ac = 0</math> (multiply by 4a)
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| − | :<math>4a^2x^2 + 4abx = -4ac</math> (subtract 4ac)
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| − | :<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)
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| − | :<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.
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| − | [[category:mathematics]]
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| − | [[category:algebra]]
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