How this instrument works
Completing the square works, in the sense of always producing a valid vertex form a(x−p)²+q, for absolutely any quadratic ax²+bx+c — that mechanical rewriting never fails. What varies is what happens NEXT: setting that vertex form to zero and solving requires taking a square root, and whether that square root lands on a real number depends entirely on the sign of one quantity, the discriminant, b²−4ac.
A positive discriminant means the completed-square form yields two distinct real roots once solved. A discriminant of exactly zero means it yields exactly one repeated real root — the vertex itself touches zero. A negative discriminant means completing the square still produces a perfectly valid vertex form, but solving it would require the square root of a negative number, meaning the quadratic has no real roots at all, only complex ones.
This page's companion, Completing the Square Practice, performs the actual vertex-form transformation itself (finding p and q); this page instead answers the narrower question of what kind of roots that transformation is heading toward, before the full transformation is even carried out.
- Enter the quadratic's three coefficients into the a, b, and c fields.
- Read Discriminant: the value b²−4ac.
- A positive result means two distinct real roots exist; zero means one repeated real root; negative means no real roots at all.
- Cross-check against the Completing the Square Practice page on this site to see the full vertex form these same coefficients produce.
Worked example — testing three quadratics
x²−6x+5 (a=1, b=−6, c=5) has a discriminant of (−6)²−4×1×5=36−20=16, positive — completing the square here does lead to two real roots (in fact 5 and 1), since the vertex form eventually needs a square root of a positive number.
x²+2x+5 (a=1, b=2, c=5) has a discriminant of 4−20=−16, negative — completing the square still gives a valid vertex form, (x+1)²+4, but solving it would need the square root of −4, meaning this quadratic has no real roots. x²−4x+4 (a=1, b=−4, c=4) sits at the boundary: a discriminant of exactly 0, giving a perfect square, (x−2)², with a single repeated real root at x=2.
Questions
Does completing the square always work?
As a mechanical rewriting into vertex form, yes, for every quadratic without exception — what varies is only whether the roots that form leads to afterward turn out to be real or complex, which depends on the sign of the discriminant.
What does a negative discriminant mean?
That the quadratic has no real roots — completing the square still produces a valid vertex form, but solving it for x would require taking the square root of a negative number, landing on complex roots instead of real ones.
What does a discriminant of exactly zero mean?
That the quadratic has exactly one repeated real root — its vertex form is a perfect square touching zero at a single point, rather than crossing through it at two distinct locations.
How is this different from the Completing the Square Practice page on this site?
That page performs the full vertex-form transformation, finding the actual p and q values; this page instead checks just the discriminant, answering only whether real roots exist before doing the fuller transformation.
Is the discriminant the same as the vertex y-coordinate?
No — they're related but distinct: the discriminant is b²−4ac, while the vertex y-coordinate is c−b²⁄4a. Both depend on the same three coefficients, but they answer different questions (root existence versus the vertex's actual height).