Same Math › Algebra 1 · grades 8–10 › Completing the square with tiles
What does completing the square mean?
Completing the square is literal. With tiles, x² + 6x is an x² square plus six x strips. Split the strips into two groups of three, one on each side, and the picture is a square missing its corner: 3 × 3 = 9 small tiles. Add the 9 and you have (x + 3)². The rule kids learn is: take half of b and square it.
- Half of b goes on each side of the x² tile.
- The missing corner is (half of b)².
- Add the corner to both sides of an equation to keep it balanced.
Why schools teach it
It's the source of the quadratic formula
Algebra 1 students complete the square to find the maximum or minimum of a quadratic (HSA.SSE.B.3b) and to solve equations, and it's how the quadratic formula is derived (HSA.REI.B.4a).
Most of us memorized “half of b, squared” with no idea why. The tiles make the reason visible, so kids can rebuild the rule when they forget it.
At the kitchen table
What to say, and what to skip
Say this
What's half of b?
What's the missing corner?
Half of b, squared.What did you add? Did you add it to the other side too?
Now it's (x + something)² = a number. What's next?
Take the square root of both sides, with ±.
Skip this
- Adding the corner to one side only.An equation stays balanced only if both sides get it.
- Forgetting the ± when taking the square root.(x + 3)² = 16 means x + 3 = 4 or x + 3 = −4.
- Squaring b instead of half of b.For 6x the corner is 9, not 36.
See it in action
Solving x² + 8x + 7 = 0
Move the 7 over
x² + 8x = −7Half of 8 is 4, and 4² = 16. Add 16 to both sides
x² + 8x + 16 = 9Write the square
(x + 4)² = 9Square root with ±, then solve
x + 4 = ±3, so x = −1 or x = −7
If a isn't 1, divide everything by a first.
Free practice sheet
Make a completing the square sheet
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Parents also ask
Why learn this if factoring is faster?
It works on every quadratic, even ones that don't factor, and it finds the vertex of a parabola.
What if b is odd?
Half of b is a fraction. For x² + 5x, the corner is (5/2)² = 25/4. The tiles get awkward, but the rule is the same.
What if there's a number in front of x²?
Divide every term by it first, so x² stands alone.
How does this lead to the quadratic formula?
Completing the square on ax² + bx + c = 0 with letters instead of numbers gives the formula.