Eleventh Grade · Math · week 10

Completing the square

Solve quadratic equations, and convert standard form to vertex form, by completing the square.

Not every quadratic factors nicely with integers, so we need a method that always works. Completing the square turns any quadratic into a perfect square trinomial by adding (and balancing) exactly the right constant -- and a perfect square trinomial takes just one square root to solve.

To complete the square for x^2+bx, take half of b, square it, and add that number -- (b/2)^2 always makes x^2+bx+(b/2)^2 factor as (x+b/2)^2. Add that same number to BOTH sides of an equation (or add and immediately subtract it, when rewriting a function), then take the square root of both sides -- remembering both the positive AND negative root.

📋 For the teacher
Teach it (10–15 min):
  • Write the phrase "half, then square" on an index card -- it is the entire method in three words, and it never changes.
  • Show why adding the SAME number to both sides is legal -- it is the exact same balance-scale idea from solving equations back in Algebra I.
  • Make the plus-or-minus step non-negotiable: have them write it explicitly every time, even when it feels obvious, since skipping it loses a whole solution.
  • Use this method to preview WHY the quadratic formula (next week) works -- completing the square on ax^2+bx+c=0 in general IS the derivation.
Talk about it:
  • Why do you add (b/2)^2, specifically, to complete the square?
  • Why must you take both the positive and negative square root?
  • How does completing the square connect to converting standard form into vertex form?

Mastery looks like: They complete the square to solve a quadratic (keeping both roots) and to convert standard form to vertex form.

Common stumbles: Adding (b/2)^2 to only one side of the equation; dropping the negative root when taking a square root.

Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
Records attendance + 15 min of Math for this learner.
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