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Completing the Square Calculator

Rewrite ax² + bx + c into vertex form a(x − h)² + k, with the vertex, the completed-square steps and the real or complex roots.

Related tools

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Open Completing the Square Calculator and you get an answer straight away, with no account to create. Its place is under Algebra & equations; Square root calculator and Chi-Square Test Calculator answer the questions closest to this one.

How to use it

  1. Open the tool — no signup or install needed.
  2. Enter your input or adjust the available options.
  3. Get your result instantly, then copy or download it.

Frequently asked questions

What does Completing the Square Calculator do?

Rewrite ax² + bx + c into vertex form a(x − h)² + k, with the vertex, the completed-square steps and the real or complex roots.

When would I actually use this?

Getting through a problem set: solving for the unknown, factoring an expression, and — more usefully — seeing the steps that got there.

What is the most common mistake?

Losing a solution when both sides are squared or divided by an expression. Squaring can add roots that do not satisfy the original equation, and dividing can remove one — check every answer back in the original.

How is Completing the Square Calculator different from Square root calculator?

They sit next to each other but answer different questions: Square root calculator is the one to open when you need it to compute the square root of a number, plus its square. Pick whichever matches what you're starting from — both are free.

Is there a tool for the next step?

Chi-Square Test Calculator is the closest one after this: χ² test of independence on any contingency table, showing observed, expected, the contribution and the standardised residual of every cell, plus Cramér's V. Cells with an expected count below 5 are flagged, because that is when the test stops being valid.

What else is worth having open alongside it?

Root Mean Square Calculator and Magic square generator — they come up in the same task often enough to be worth a second tab.

Where do the figures come from?

The algebra is symbolic where the expression allows it and numeric otherwise, and the tool says which it used. Exact results are given as fractions or radicals rather than rounded decimals.

Further reading

All guides →
How-toHow to Use the Quadratic Formula: Solving ax² + bx + c = 0Solve any quadratic equation with x = (−b ± √(b² − 4ac)) / 2a, and read the discriminant to know how many real solutions to expect.How-toHow to Find a Cube Root by HandA step-by-step guide to finding a cube root by estimation and refinement — plus how it differs from a square root and when a value stays irrational.ExplainerHow Exponents Work: Powers and the Laws of ExponentsAn exponent is repeated multiplication. Learn the laws of exponents, what negative and fractional powers mean, and why anything to the zero is one.How-toHow to Calculate the Square Metres of a RoomFind a room's area by multiplying length by width. Here's the method for rectangular, L-shaped and round rooms, and why area drives your material orders.ExplainerThe Flat Pattern of a Cone Is a Sector, Not a CircleRoll a 50 mm cone 80 mm tall and its pattern is a 190.8° slice of a 94 mm circle. Flatten a nearly cylindrical lampshade and the pattern radius runs to 839 mm — the shallower the taper, the further away the apex.ExplainerA 1 % Drop Rate Does Not Mean One Hundred RunsAt 1 %, a hundred attempts give you 63.4 % — not certainty. Ninety per cent takes 230 attempts and ninety-nine takes 459, and more than a third of players are still empty-handed at a hundred.