Enter the coefficients a, b, and c of a quadratic equation (ax² + bx + c = 0) and this Completing the Square Calculator walks you through the full solution. You get the vertex form, the discriminant, and both roots (x₁ and x₂) — including complex roots when the discriminant is negative.
Vertex Form
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Discriminant (b² − 4ac)
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Root x₁
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Root x₂
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Vertex x (axis of symmetry)
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Vertex y (minimum/maximum)
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Nature of Roots
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Completing the square is an algebraic method that rewrites a quadratic equation ax² + bx + c = 0 into the form a(x + h)² + k = 0 (vertex form). This makes it easy to identify the vertex of the parabola and to solve for x, even when the equation cannot be factored easily.
First, divide through by a so the x² coefficient is 1. Move the constant to the right side. Take half of the b coefficient, square it, and add it to both sides. The left side becomes a perfect square trinomial (x + b/2)². Finally, take the square root of both sides and solve for x.
When a ≠ 1, divide the entire equation by a first to make the leading coefficient 1. Then proceed with completing the square as normal. The calculator handles this automatically and rescales all steps accordingly.
When b = 0, the equation simplifies to ax² + c = 0. There is no middle term to complete, so you simply solve x² = −c/a directly. The calculator still displays the vertex form a(x + 0)² + c and solves for x.
The discriminant is b² − 4ac. If it is positive, the equation has two distinct real roots. If it equals zero, there is exactly one real root (a repeated root). If it is negative, the equation has two complex (imaginary) roots.
Yes. When the discriminant is negative, the square root of a negative number produces imaginary values. The roots take the form x = h ± ki, where i is the imaginary unit. The calculator identifies this case and reports the nature of the roots.
Vertex form is a(x − h)² + k = 0, where (h, k) is the vertex of the parabola. Here h = −b/(2a) and k = c − b²/(4a). Completing the square is essentially the process of converting standard form into vertex form.
Completing the square reveals the vertex and structure of the parabola, which is valuable in graphing and optimization problems. The quadratic formula is actually derived from completing the square, so both methods always give the same roots — but completing the square provides more geometric insight.