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added background
It is first taught using equations with "friendly" numbers in place of ''a'', ''b'', and ''c'' to get the student used to the process.
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]]:
By then applying the process to the general equation, we can derive the [[quadratic formula]]:
<math>ax^2 + bx + c = 0</math> (given)
:<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 = 0</math> (multiply by 4a)
<math>4a^2x^2 + 4abx = -4ac</math> (subtract 4ac)
:<math>4a^2x^2 + 4abx = -4ac</math> (subtract 4ac)
<math>4a^2x^2 + 4abx + b^2 = -4ac + b^2</math> (add b^2)
:<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)^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> (take sqaure root, allow for both roots)
<math>2ax = -b \pm \sqrt{b^2 - 4ac}</math> (subtract b)
:<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)
:<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.
We can now determine the real or imaginary roots of any quadratic equation by simply inserting ''a'', ''b'', and ''c'' into the formula.