Completing the square
Completing the square is a method for solving for the roots of the general quadratic equation:
ax2 + bx + c = 0, 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:
- d2x2 + 2dex + e2 = f where d, e and f are constants expressed in a, b, and c.
- This resolves to:
- (dx + e)2 = f (grouping)
- (take square root)
- (subtract e)
- (divide by e)
By then applying the process to the general equation, we can derive the quadratic formula:
- ax2 + bx + c = 0 (given)
- 4a2x2 + 4abx + 4ac = 0 (multiply by 4a)
- 4a2x2 + 4abx = − 4ac (subtract 4ac)
- 4a2x2 + 4abx + b2 = − 4ac + b2 (add b^2)
- (2ax + b)2 = b2 − 4ac (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.