Ninth Grade · Math · week 28

Completing the square

Solve quadratic equations by completing the square, including equations with a leading coefficient other than 1.

Not every quadratic factors nicely, so algebra needs a method that always works. Completing the square builds a perfect square trinomial on purpose: for x² + bx, adding (b/2)² creates exactly x² + bx + (b/2)², which factors instantly as (x + b/2)². The trick is doing this to an EQUATION - whatever you add to one side, you must add to the other, keeping the balance exactly as it was in term one.

The full method: move the constant to the other side, add (b/2)² to both sides, rewrite the left side as a squared binomial, then take the square root of both sides - remembering both the positive AND negative root - and finish solving for x. If the leading coefficient is not 1, divide every term by it first, since the (b/2)² pattern only works cleanly when the x² term has coefficient 1.

📋 For the grown-up teacher
Teach it (10–15 min):
  • Show the "adding a physical square" picture at least once - a literal square drawn on graph paper with side (x + b/2) - so "completing the square" means something geometric, not just symbolic.
  • Make "same thing to both sides" the spoken rule every single time a number gets added - this is term one's balance-scale idea again, in a new costume.
  • The ± after taking a square root is the single easiest thing to forget - build a pause into the routine specifically for it.
  • When a ≠ 1, physically divide every term (including the constant) by a before doing anything else, and say why: the pattern needs a clean x² first.
Talk about it:
  • Why do you add (b/2)² specifically, instead of any other number?
  • What happens to a solution if you forget the negative square root?
  • Why must you divide by the leading coefficient first when a is not 1?

Mastery looks like: They complete the square correctly (including the ± step) to solve a quadratic, and divide by a first when the leading coefficient is not 1.

Common stumbles: Forgetting the negative root after taking a square root; adding (b/2)² to only one side of the equation.

Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
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