The formula this page uses
x² + bx → (x + b/2)² − (b/2)² a x² + bx + c → a(x + b/(2a))² + c − b²/(4a)
Complete the square on any quadratic
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Algebra
Completing the Square Calculator explained
The short version
- Completing the square rewrites x² + bx as a single squared bracket plus a leftover number.
- The number you add is always half of b, then squared. Add it, and subtract it straight back so the value does not change.
- If the leading number in front of x² is not 1, pull it out before you halve anything, or every later step is off.
What each part means
SymbolWhat it means
b — Coefficient of xThe number in front of the single x, with its sign. Half of it goes inside the bracket.
b/2 — Half of bThis exact number appears inside the finished bracket, sign and all.
(b/2)² — The completing numberWhat you add and subtract. It is never negative, because it is something squared.
a — Coefficient of x²If a is not 1, factor it out of the x² and x terms first. It stays outside the bracket.
Show your work: a full example
- Start with the equation3x² − 12x + 5 = 0
- Factor the 3 out of the first two terms only3(x² − 4x) + 5 = 0
- Half of −4 is −2, and (−2)² is 43(x² − 4x + 4 − 4) + 5 = 0
- Pull the spare −4 out through the 3, so it becomes −123(x − 2)² − 12 + 5 = 0
- Tidy the constants3(x − 2)² = 7
- Divide by 3, then take both square roots(x − 2)² = 7/3, so x − 2 = ±1.5275
- Write the two answersx = 3.5275 or x = 0.4725
A second, different case
- A different case: the leftover goes negativex² + 4x + 9 = 0
- Move the constant acrossx² + 4x = −9
- Half of 4 is 2, and 2² is 4, so add 4 to both sidesx² + 4x + 4 = −9 + 4
- Fold the left side into one bracket(x + 2)² = −5
- Read the right sideno real number squares to −5, so there is no real solution
- Carry on into the complex numbersx + 2 = ±i√5, so x = −2 ± 2.2361i
Copy-ready example
3x² − 12x + 5 = 0
Start with the equation
Questions about the Completing the Square Calculator
Why do I have to factor a out before halving b?
Because the rule 'half of b, squared' was built for x² + bx, where the x² has nothing in front of it. In 3x² − 12x the b you see is −12, but the b that matters inside the bracket is −4, which only appears after the 3 comes out.
What if b is an odd number?
Nothing breaks, you just get a fraction. For x² + 5x, half of 5 is 2.5 and 2.5² is 6.25, so the bracket is (x + 2.5)². Many people keep it as (x + 5/2)² and add 25/4 instead, which avoids decimals entirely.
Is completing the square the same thing as the quadratic formula?
The formula is what you get if you complete the square on a general ax² + bx + c once and remember the result. That is why the 2a in the formula's denominator exists: it is the b/(2a) from this method, still sitting there.
When is completing the square worth the extra work?
When you want the turning point rather than the roots. 3(x − 2)² − 7 tells you instantly that the lowest point is at (2, −7). The formula would hand you 3.5275 and 0.4725 and leave you to average them.
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