Completing the Square Calculator

Completing the square is the algebraic technique of rewriting a quadratic $ax^2 + bx + c$ as:

$a(x - h)^2 + k$

where $(h, k)$ is the vertex of the parabola.

Why this matters:

• Reveals the vertex (minimum/maximum point) of a parabola at a glance.
• Lets you solve any quadratic equation without the quadratic formula.
• Is the underlying technique that derives the quadratic formula.
• Used to evaluate $\int \frac{1}{x^2 + bx + c}\,dx$ in calculus (reduces to arctan).
• Essential for understanding Gaussian integrals and many topics in physics.

The core identity that makes it work:

$x^2 + bx + \left(\frac{b}{2}\right)^2 = \left(x + \frac{b}{2}\right)^2$

Case 1: Leading Coefficient is 1

For $x^2 + bx + c$:

1. Take half of $b$ and square it: $(b/2)^2$.
2. Add and subtract this quantity: $x^2 + bx + (b/2)^2 - (b/2)^2 + c$.
3. Group the perfect square: $(x + b/2)^2 + c - (b/2)^2$.

Example: $x^2 + 6x + 5$

• Half of 6 is 3. Squared: 9.
• $x^2 + 6x + 9 - 9 + 5 = (x + 3)^2 - 4$

Vertex form: $(x + 3)^2 - 4$, vertex at $(-3, -4)$.

Case 2: Leading Coefficient is Not 1

For $ax^2 + bx + c$, $a \neq 1$:

1. Factor $a$ out of the first two terms: $a\left(x^2 + \frac{b}{a}x\right) + c$.
2. Complete the square inside the parentheses: half of $b/a$ is $b/(2a)$, squared is $b^2/(4a^2)$.
3. Add and subtract inside: $a\left(x^2 + \frac{b}{a}x + \frac{b^2}{4a^2}\right) - a \cdot \frac{b^2}{4a^2} + c$.
4. Simplify: $a\left(x + \frac{b}{2a}\right)^2 + c - \frac{b^2}{4a}$.

Note that when you 'undo' the added term, you multiply by $a$ since the inside is multiplied by $a$.

Solving a Quadratic Equation

For $ax^2 + bx + c = 0$:

1. Complete the square to get $a(x - h)^2 + k = 0$.
2. Isolate the squared term: $(x - h)^2 = -k/a$.
3. Take square roots: $x - h = \pm\sqrt{-k/a}$.
4. Solve: $x = h \pm \sqrt{-k/a}$.

This is essentially what the quadratic formula does in a single compact expression.

• Forgetting to balance: When you add $(b/2)^2$, you must subtract it too. Otherwise you've changed the expression.
• Wrong coefficient handling: If $a \neq 1$, you must factor $a$ out of the first two terms before completing the square, then multiply your correction by $a$ when distributing back.
• Sign errors with $\pm$: After taking square roots, both branches must be kept. Dropping the $\pm$ loses a solution.
• Half of $b$ vs $b/2a$: When the leading coefficient is 1, take half of $b$. When it's not, factor first — then take half of the new coefficient.
• Forgetting to simplify the constant: After completing the square, combine the leftover constants into a single $k$.