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'''Completing the square''' is a method for solving for the roots of the general [[quadratic equation]]:
<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.
By then applying the process to the general equation, we can derive the [[quadratic formula]]:
<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)
<math>(2ax + b)^2 = b^2 - 4ac</math> (group each side)
<math>2ax + b = \pm \sqrt{b^2 - 4ac}</math> (take sqaure root, allow for both roots)
<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)
We can now determine the real or imaginary roots of any quadratic equation by simply inserting ''a'', ''b'', and ''c'' into the formula.
<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.
By then applying the process to the general equation, we can derive the [[quadratic formula]]:
<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)
<math>(2ax + b)^2 = b^2 - 4ac</math> (group each side)
<math>2ax + b = \pm \sqrt{b^2 - 4ac}</math> (take sqaure root, allow for both roots)
<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)
We can now determine the real or imaginary roots of any quadratic equation by simply inserting ''a'', ''b'', and ''c'' into the formula.