Completing the Square: Geometry of Algebra

What is completing the square?

Completing the square rewrites $x^2 + bx + c$ as $(x + h)^2 + k$ — the vertex form. This reveals the vertex of the parabola and makes it easy to solve the equation.

Step-by-step: Transform $x^2 + 6x + 5$ into vertex form

Step 1 — Focus on $x^2 + 6x$. Take half the coefficient of $x$: $6/2 = 3$. Square it: $3^2 = 9$.

Step 2 — Add and subtract 9 (adding zero in a clever way):

$x^2 + 6x + 9 - 9 + 5$

Step 3 — The first three terms form a perfect square:

$(x + 3)^2 - 9 + 5 = (x + 3)^2 - 4$

Result: $x^2 + 6x + 5 = (x + 3)^2 - 4$.

Vertex: $(-3, -4)$. The parabola opens upward with minimum at $y = -4$.

Check: $(x+3)^2 - 4 = x^2 + 6x + 9 - 4 = x^2 + 6x + 5$ ✓

The geometric picture

$x^2 + 6x$ is an $x$-by-$x$ square plus a $6$-by-$x$ rectangle. Split the rectangle into two $3$-by-$x$ pieces and attach to two sides of the square → you get an $(x+3)$-by-$(x+3)$ square with a $3 \times 3$ corner missing. Adding 9 fills the corner.

Why learn this?

Completing the square is used to: (1) derive the quadratic formula, (2) find the vertex of a parabola, (3) rewrite conic equations in standard form, (4) integrate certain expressions in calculus.

Try it in the visualization

Adjust $b$ and $c$. The algebraic steps animate. The parabola shifts to show the vertex form. The geometric square-building visualization shows how the rectangle becomes a perfect square.