This completing the square calculator rewrites a quadratic ax² + bx + c in vertex form a(x − h)² + k, reports the vertex and the roots — the transformation behind the quadratic formula itself.
Standard Form Hides It; Vertex Form Shows It
ax² + bx + c tells you the y-intercept and little else. Completing the square repackages the same parabola as a(x − h)² + k, where (h, k) is the vertex — minimum or maximum visible at a glance, plus the axis of symmetry x = h. Our Quadratic Formula Calculator is literally this procedure, pre-computed for all a, b, c. Part of our Math Calculators collection.
Worked Example (Verified)
- x² + 6x + 5 (defaults): half of 6 is 3, 3² = 9 → (x + 3)² − 9 + 5 = (x + 3)² − 4 · vertex (−3, −4) · roots: (x+3)² = 4 → x = −1 and −5
- Non-monic: 2x² + 8x + 6 → 2(x + 2)² − 2 (factor the 2 first, complete inside), vertex (−2, −2)
- The no-real-roots case looks identical in vertex form (e.g. (x+1)² + 3): the +3 floor is exactly why the parabola never reaches the axis — the discriminant’s story told geometrically
- Optimization payoff: a projectile/revenue parabola’s peak is the vertex — completing the square is calculus-free optimization
Where Else This Move Appears
- Circle equations: x² + y² + Dx + Ey + F = 0 becomes center/radius form by completing the square twice.
- Calculus: integrals of 1/(quadratic) start by completing the square toward arctan forms.
- Statistics: the normal distribution’s algebra and least squares both run on this exact maneuver.
- Deriving the quadratic formula in class — you complete the square once, generally, and never divide polynomials again.
Related Tools
- Quadratic Formula Calculator — the formula this method proves
- Factorial Calculator — more algebra essentials
- Math Calculators — the full collection
Frequently Asked Questions
How do you complete the square?
For x² + bx + c: halve b, square the result, add and subtract it: x² + 6x + 5 → (x + 3)² − 9 + 5 = (x + 3)² − 4. With a leading a ≠ 1, factor a out of the x-terms first. This calculator does it for any a, b, c and reports vertex and roots.
What is vertex form?
a(x − h)² + k — the quadratic rewritten around its turning point (h, k). Sign care: (x + 3)² means h = −3 (the form subtracts h). The vertex is the minimum when a > 0 and the maximum when a < 0.
Why complete the square instead of using the formula?
Because sometimes you need the vertex, not the roots — optimization, graph sketching, circle equations, calculus prep. And pedagogically the formula is completing the square done once in general; knowing the method means the formula can be re-derived, not just recalled.
What if the roots aren’t real?
Vertex form shows why: if a and k have the same sign (e.g. (x+1)² + 3 with a = 1, k = 3), the parabola never touches the x-axis. The calculator reports the complex pair (h ± imaginary part) in that case rather than pretending nothing exists.
Does completing the square work with fractions?
Yes — odd b values produce halves (b = 5 → (x + 2.5)²), and fractions stay exact in the (b/2a)² arithmetic. The calculator rounds display to 4 decimals; the method itself is exact.
Is this completing the square calculator free?
Yes — free, instant, no sign-up.