Completing the Square Calculator

What completing the square means

To complete the square is to rewrite a quadratic $ax^2 + bx + c$ so its variable terms form a single squared binomial: $a(x - h)^2 + k$. This is called vertex form, because $(h, k)$ is the vertex of the parabola $y = ax^2 + bx + c$. Once the equation is in this shape, solving $a(x-h)^2 + k = 0$ for $x$ is just arithmetic — no factoring required.

The 9-step method

1. Write the equation as $ax^2 + bx + c = 0$.
2. If $a \neq 1$, divide every term by $a$.
3. Move the constant term to the right side.
4. Take half of the $x$-coefficient and square it: $(b/2)^2$.
5. Add that value to both sides.
6. Write the left side as a perfect square, $(x + b/2)^2$.
7. Take the square root of both sides — remember $\pm$.
8. Move the remaining constant to isolate $x$.
9. Simplify: the $\pm$ gives the two roots.

Worked example — a ≠ 1

Solve $2x^2 - 12x + 7 = 0$.

1. Divide by $a = 2$: $x^2 - 6x + 3.5 = 0$.
2. Move the constant: $x^2 - 6x = -3.5$.
3. Half the $x$-coefficient, squared: $(-6/2)^2 = 9$.
4. Add 9 to both sides: $x^2 - 6x + 9 = -3.5 + 9 = 5.5$.
5. Perfect square: $(x - 3)^2 = 5.5$.
6. Square root both sides: $x - 3 = \pm\sqrt{5.5}$.
7. Isolate $x$: $x = 3 \pm \sqrt{5.5} \approx 5.3452$ or $0.6548$.

The vertex form is $2(x - 3)^2 - 11$, so the vertex of this parabola is $(3, -11)$ — and it matches the roots you would get from the Quadratic Formula calculator for the same equation.

Worked example — no linear term

Solve $x^2 - 4 = 0$. There is no $x$ term, so steps 3–6 are trivial: $x^2 = 4$, so $x = \pm\sqrt{4} = \pm 2$. Vertex form is simply $x^2 - 4$, with vertex $(0, -4)$.

Why (b/2)² completes the square

Expand $(x + b/2)^2 = x^2 + bx + (b/2)^2$. Compare that to $x^2 + bx$ — the only thing missing is the constant $(b/2)^2$. Adding it turns the expression into a perfect square, which is exactly why the method is called "completing" the square.

Common mistakes

• Forgetting to divide by $a$ first when $a \neq 1$.
• Adding $(b/2)^2$ to only one side — it must go on both sides to keep the equation balanced.
• Squaring before halving — it is $(b/2)^2$, not $b^2/2$.
• Dropping the $\pm$ when taking the square root, which loses one of the two roots.

Vertex form and the quadratic formula

Vertex form $a(x - h)^2 + k$ directly gives the vertex $(h, k)$ of the parabola — useful for graphing, and for optimization problems (maximum height, minimum cost, and so on). It is also where the quadratic formula comes from: complete the square on the general equation $ax^2 + bx + c = 0$ and solving for $x$ reproduces the formula exactly, with $h = -b/(2a)$ and $k = c - b^2/(4a)$.

Where this shows up

Completing the square is the standard route to a parabola's vertex — the highest or lowest point of anything modeled by a quadratic: projectile height, profit curves, or the minimum of a cost function. It is also the technique behind deriving the general formula for solving any quadratic equation.