Completing the Square

Lesson 10 of 12 · 7 min · Free, no account needed

What you will learn

Completing the square: split the strips in half, fill the corner, and owe back what you added. x² + 6x + 4 = (x + 3)² − 5.
The completed form shows the vertex: (x + 3)² − 5 has its lowest point at (−3, −5).
Undoing a square gives two answers: (x + 3)² = 5 means x + 3 = √5 or x + 3 = −√5.

x² + 6x + 4 won't fold into a rectangle. Try anyway.
Drag the 6 strips to the right or below. Try at least three different splits.
Make a square instead. Buy what's missing.
Put 3 strips on each side. The corner has room for 9 and you have 4: buy the rest from the shop, and owe it back.
Where does y = x² + 6x + 4 sit?
Turn h and k to move y = (x − h)² + k onto the dashed curve.
That's completing the square.
Completing the square: halve the strips, fill the corner, owe back what you bought. x² + 6x + 4 = (x + 3)² − 5.
The completed form is the vertex form: the lowest point is at x = −3, y = −5.
To solve x² + 6x + 4 = 0, write (x + 3)² = 5. Undo the square with a square root, and remember both signs: x + 3 = ±√5.
(x + 3)² = 5. Where does the curve reach the level 5?
Drag both pins to where the curve meets the dashed line.
Game: vertex hunt.
Complete the square in your head and drop the flag on the vertex. Then the curve appears.
Build your own curve that dips below the ground.
Slide the strips b and the units c. The square completes itself.
The ball from last lesson: h = 20t − 5t². How high does it go?
Write it as −5(t² − 4t) and complete the square inside the bracket.
Speed round: tap each vertex.
Three curves. The last one is yours.

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