Why That Number?

Where it usually goes wrong

"Add the square of half the middle coefficient" gets memorized as a step in a procedure, disconnected from anything, so a child who mixes it up — halving after squaring, or squaring the whole coefficient instead of half of it — has no picture to check the answer against. The number is not arbitrary: it is the area of an actual rectangle, sized to fit an actual gap, and it can be measured rather than recalled.

Try this

• Drag the strip out to width 3, so b is 6. Read the open corner: 3 by 3, not 6 by 6 — it is half the strip, not the whole one.
• Press "Fill the corner" and check the number against half of b, squared. They match because the corner IS that square, not merely equal to it.
• Drag the strip back to width 1, so b is 2. The corner shrinks to 1 by 1 — completing the square never adds much when b is small.

Where this goes next

This is the same rectangle the quadratic formula's own derivation completes, and it is why √(b² − 4ac) has a 4 in it: the formula is this picture, finished algebraically instead of by drawing it.