Completing the Square Calculator with Steps

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Formula at a glance

Completing the Square Calculator: Completed square — ax² + bx + c = a(x + b/2a)² + (c − b²/4a). Example: x² + 6x + 5 → Completed square: x² + 6x + 5 = (x + 3)² − 4 — How the Completing the Square Calculator works: Completed square — ax² + bx + c = a(x + b/2a)² + (c − b²/4a)

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📘 About the Completing the Square Calculator

Completing the square rewrites a quadratic as a perfect square plus a constant: x² + 6x + 5 = (x + 3)² − 4. It reveals the vertex of the parabola, solves equations, and is how the quadratic formula is derived. It is also used to find the centre of circles and to integrate.

Enter a, b and c (fractions allowed). The calculator factors out a, halves the x-coefficient, adds and subtracts its square, and writes the perfect square — then, if you choose, isolates the square and takes square roots to solve the equation, keeping exact fractions and surds.

🧮 Formula

Completed square

ax² + bx + c = a(x + b/2a)² + (c − b²/4a)

Key step

add and subtract (b/2a)²

✏️ Worked examples

x² + 6x + 5

You enter
a	1
b	6
c	5
Also solve ax² + bx + c = 0?	Yes

You get
Completed square	x² + 6x + 5 = (x + 3)² − 4
Vertex	(−3, −4)
Form	a(x − h)² + k with a = 1, h = −3, k = −4
Solutions of = 0	x = −5, x = −1

x² + 6x + 5 = (x + 3)² − 4.

2x² + 8x + 5

You enter
a	2
b	8
c	5
Also solve ax² + bx + c = 0?	Yes

You get
Completed square	2x² + 8x + 5 = 2(x + 2)² − 3
Vertex	(−2, −3)
Form	a(x − h)² + k with a = 2, h = −2, k = −3
Solutions of = 0	x = −√6/2 − 2, x = √6/2 − 2 (≈ −3.22474, −0.775255)

2x² + 8x + 5 = 2(x + 2)² − 3.

🧭 How to use this calculator

What you enter

a
b
c
Also solve ax² + bx + c = 0?

What you get

Completed square
Vertex
Form
Solutions of = 0

❓ Frequently asked questions

How do you complete the square?

For x² + bx: add (b/2)² to make a perfect square, (x + b/2)², and subtract it again to keep the value: x² + 6x = (x + 3)² − 9.

What if a is not 1?

Factor a out of the x² and x terms first: 2x² + 8x + 5 = 2(x² + 4x) + 5 = 2(x + 2)² − 3.

Why complete the square instead of using the formula?

It shows the vertex directly and works with letters as coefficients. The quadratic formula itself comes from completing the square on ax² + bx + c.

□ Completing the Square Calculator