Completing the square, step by step. Enter a, b and c for any quadratic and see the vertex form, the vertex (h, k), an area-model diagram, and the parabola graphed.
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Completing the square rewrites a quadratic ax² + bx + c in vertex form a(x − h)² + k, where (h, k) is the vertex of the parabola. Enter the three coefficients and this tool derives h = −b/(2a) and k = c − b²/(4a), shows the factoring step when the leading coefficient is not 1, and narrates every intermediate rewrite.
The area model makes the algebra visual: the x² term is drawn as a square of side x, the bx term splits into two rectangles of width b/(2a), and the missing corner is the small square of area (b/(2a))² that literally completes the larger square. The vertex is also plotted on the graphed parabola.
Completing the square rewrites ax² + bx + c as a(x − h)² + k, called vertex form. That form shows the parabola's vertex (h, k) directly and makes the quadratic easy to solve by taking square roots.
Factor the leading coefficient out of the x² and x terms first, complete the square inside the parentheses, then distribute. For example, 2x² + 8x + 3 becomes 2(x² + 4x) + 3, then 2[(x + 2)² − 4] + 3, and finally 2(x + 2)² − 5.
The area model draws x² as a square of side x and splits bx into two rectangles of width b/(2a) laid against two of its sides. The missing corner is a small square of area (b/(2a))², which is exactly what you add to complete the square.
The vertex x-coordinate is h = −b/(2a) and the vertex y-coordinate is k = c − b²/(4a). Both fall straight out of the completed-square rewrite, so the vertex never has to be found by graphing.
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