Mathematics › 🌆 Grade 9 › Square Completer: Completing the Square

Square Completer: Completing the Square

Every near-square shape is missing exactly one corner tile — add it back, and the trinomial becomes a perfect square.

Grade 9Algebra 1

Square Completer: Completing the Square — illustration

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The big idea: Completing the square turns an expression like x² + bx into a true perfect square, (x + b÷2)², by adding the one missing piece: (b÷2)². Picture x² + bx as an almost-square tile arrangement with a small corner missing — completing the square is literally filling that gap. This trick lets you rewrite any quadratic in vertex form and read off its vertex directly.

🎯 By the end, you'll be able to

Explain completing the square as filling in a missing corner tile geometrically
Complete the square for an expression x² + bx by adding (b÷2)²
Convert a quadratic from standard form to vertex form by completing the square
Read the vertex of a quadratic directly from its completed-square (vertex) form

📎 You should already know

Multiplying binomials (FOIL)
Vertex form of a quadratic

A square missing one corner

Take x² + bx and build it out of tiles: one big x-by-x square, and b thin x-by-1 strips laid along two of its sides. Arrange the strips evenly along the top and the right side, and you get something that is almost a bigger square — except for one small gap in the corner.

That missing corner has a precise size: (b÷2) by (b÷2). Add that one small tile in, and the whole shape becomes a genuine square with side length x + b÷2.

🔑 Add (b÷2)² to complete the square

x² + bx + (b÷2)² is a perfect square: it equals (x + b÷2)². The number you add, (b÷2)², is exactly the area of the missing corner tile.

\[ x^2 + bx + \left(\dfrac{b}{2}\right)^2 = \left(x + \dfrac{b}{2}\right)^2 \]

Completing the square: add half of b, squared, and the trinomial becomes a perfect square binomial.

🎮 Square Completer LIVE

Add the missing corner tile to complete the square and reveal vertex form.

From standard form to vertex form

To rewrite y = x² + bx + c in vertex form, complete the square on the x² + bx part. Since you cannot add a number to one side of an equation without balancing it, you add (b÷2)² and then immediately subtract it again, so the value of the expression never actually changes.

📝 Worked example: Complete the square for x² + 6x.

Take half of 6: 6 ÷ 2 = 3.
Square it: 3² = 9.
Add 9: x² + 6x + 9 factors as a perfect square.

✓ x² + 6x + 9 = (x + 3)².

📝 Worked example: Convert y = x² + 8x + 10 to vertex form.

Half of 8 is 4, and 4² = 16, so add and subtract 16: y = x² + 8x + 16 − 16 + 10.
The first three terms form a perfect square: x² + 8x + 16 = (x + 4)².
Combine the remaining constants: −16 + 10 = −6.

✓ y = (x + 4)² − 6, with vertex (−4, −6).

⚠️ Whatever you add, you must also subtract

Adding (b÷2)² changes the expression unless you immediately subtract the same amount. Forgetting to subtract it back out is the most common mistake in completing the square — it silently changes the value of the whole expression.

✨ This is where the quadratic formula comes from

Completing the square is not just a trick for one problem — applying it to the general equation ax² + bx + c = 0 is exactly how the quadratic formula itself is derived. Every quadratic equation can be solved this same way.

Why bother completing the square?

Once a quadratic is in vertex form, its vertex — the maximum or minimum point — is visible immediately, without graphing a single point. That is often exactly the information a problem is really asking for.

Check your understanding

1. What number completes the square for x² + 10x?

Half of 10 is 5, and 5² = 25, so x² + 10x + 25 = (x + 5)².

2. Complete the square: x² − 4x + ___ is a perfect square.

Half of −4 is −2, and (−2)² = 4, so x² − 4x + 4 = (x − 2)².

3. Convert y = x² + 2x + 5 to vertex form.

Half of 2 is 1, and 1² = 1: x² + 2x + 1 − 1 + 5 = (x + 1)² + 4.

4. What is the vertex of y = (x + 1)² + 4?

In (x − h)² + k form, (x + 1)² means h = −1, and k = 4, so the vertex is (−1, 4).

5. Why must you subtract (b÷2)² right after adding it when completing the square in an expression?

Adding (b÷2)² changes the value of the expression unless it is balanced by subtracting the same amount immediately after.

✅ Key takeaways

x² + bx is an almost-square missing one corner tile of size (b÷2) by (b÷2).
Adding (b÷2)² completes the square: x² + bx + (b÷2)² = (x + b÷2)².
To convert standard form to vertex form, add and immediately subtract (b÷2)² so the value stays the same.
Once in vertex form (x − h)² + k, the vertex (h, k) is visible directly — no graphing required.
Completing the square on ax² + bx + c = 0 is exactly how the quadratic formula is derived.