Completing the Square to Solve Equations

What is "completing the square"?

Completing the square rewrites $x^2 + bx$ as a perfect square $(x + b/2)^2$ minus a correction term. This lets you solve quadratics by taking a square root instead of using the formula.

Step-by-step solution: Solve $x^2 + 6x + 2 = 0$

Step 1 — Move the constant to the right side:

$x^2 + 6x = -2$

Step 2 — Take half the coefficient of $x$, then square it. Half of 6 is 3; $3^2 = 9$. Add 9 to BOTH sides:

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

Step 3 — The left side is now a perfect square:

$(x + 3)^2 = 7$

Step 4 — Take the square root of both sides (don't forget ±):

$x + 3 = \pm\sqrt{7}$

Step 5 — Solve for $x$:

$x = -3 \pm \sqrt{7}$

$x_1 = -3 + \sqrt{7} \approx -0.354$ $x_2 = -3 - \sqrt{7} \approx -5.646$

Step 6 — Check (using $x_1$): $(-0.354)^2 + 6(-0.354) + 2 = 0.125 - 2.124 + 2 \approx 0$ ✓

The geometric picture

"Completing the square" literally means building a square. You have an $x \times x$ square and a $6 \times x$ rectangle. Cut the rectangle in half ($3 \times x$ each) and attach to two sides of the square. You get an $(x+3) \times (x+3)$ square minus the missing $3 \times 3$ corner — that's the 9 you add.

Why learn this?

Completing the square is how the quadratic formula is derived, and it's essential for putting conics in standard form, vertex form of parabolas, and integration techniques in calculus.

Try it in the visualization

Adjust $b$ and $c$. Watch the geometric square being built step by step. The parabola shifts to show the vertex form, and the roots appear on the x-axis.