Completing the Square: Deriving the Quadratic Formula

Completing the Square: Quadratic Examples
& Deriving the Quadratic Formula (page 2 of 2)

This is the original equation.	x² + 6x + 10 = 0
Move the loose number over to the other side.	x² + 6x = – 10
Take half of the coefficient on the x-term (that is, divide it by two) (don't forget the sign!), and square it. Add this square to both sides of the equation.
Convert the left-hand side to squared form. Simplify the right-hand side. Note: If you don't know about complex numbers yet, then you have to stop at this step, because a square can't equal a negative number! Otherwise, proceed...	(x + 3)² = –1
Square-root both sides. Remember to put the "±" on the right-hand side. Note the square root of a negative number!
Solve for "x =", and simplify as necessary.	x = –3 ± i

If you don't yet know about complex numbers (the numbers with "i" in them), then you would say that this quadratic has "no solution". If you do know about complexes, then you would say that this quadratic has "no real solution" or that is has a "complex solution". In either case, this quadratic had no "real" solution. Since solving "(quadratic) = 0" for x is the same as finding the x-intercepts (assuming the solutions are real numbers), it stands to reason that this quadratic should not intersect the x-axis (since x-intercepts are "real" numbers), and you can see below that it doesn't:

This relationship is always true. If you come up with a real value on the right side (a zero value is real, by the way; the square root of zero is just zero), then the quadratic will have two x-intercepts (or only one, if you get plus/minus of zero on the right side); if you get a negative on the right side, then the quadratic will not cross the x-axis.

I'll do one last "example". It has become somewhat fashionable to have students derive the Quadratic Formula themselves, by completing the square for the generic quadratic ax² + bx + c. While I can understand the impulse (showing students how the Formula was invented, and thereby giving an example of the usefulness of symbolic manipulation), the computations involved are often a bit beyond the average student at this point. Here is what the instructor is looking for:

This is the original equation.	ax² + bx + c = 0
Move the loose number to the other side.	ax² + bx = –c
Divide through by whatever is multiplied on the squared term. Take half of the x-term, and square it. Add the squared term to both sides.
Simplify on the right-hand side; in this case, simplify by converting to a common denominator.
Convert the left-hand side to square form (and do a bit more simplifying on the right).
Square-root both sides, remembering to put the "±" on the right.
Solve for "x =", and simplify as necessary.

The key to solving by completing the square is to practice, practice, practice, so you'll remember the steps when you're taking the test.

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