How this calculator works
Expand (px + q)^2 and compare coefficients: a = p^2, b = 2pq and c = q^2. For a positive a, the necessary and sufficient condition is b^2 = 4ac with c non-negative. The numerical comparison uses a relative tolerance of 1e-12. A non-negative constant is also a degenerate square. A negative leading coefficient cannot be a square of a real binomial.
Formula
(px + q)^2 = p^2*x^2 + 2pq*x + q^2. Given a > 0 and b, c = b^2/(4a) completes the square.
Worked example
4x^2 - 12x + 9 = (2x - 3)^2 because 2 x 2 x (-3) = -12. To complete x^2 + 6x + c, use c = 36/4 = 9, giving (x + 3)^2.
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Frequently asked questions
Is a zero discriminant enough?
For a genuine real quadratic square, a must also be positive. A negative multiple of a square has zero discriminant but is not itself a real square.
Can the middle term be negative?
Yes. A negative b selects opposite signs in the square-root binomial.
What does the tolerance mean?
A very small relative discrepancy can be accepted as rounding noise. Use exact symbolic arithmetic to certify nearly square decimal inputs.