Rewrite a quadratic equation in vertex form a(x − h)² + k, showing vertex, axis of symmetry, roots, and step-by-step work.
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Quadratic equation: ax² + bx + c
Vertex Form
Step-by-step
Also in Functions
Algebra
A completing-the-square calculator rewrites a quadratic expression ax² + bx + c in vertex form a(x − h)² + k. This transformation reveals the vertex of the parabola and makes it easy to identify the axis of symmetry, direction of opening, and minimum or maximum value.
Starting from ax² + bx + c, factor out a from the first two terms: a(x² + (b/a)x) + c. Then add and subtract (b/(2a))² inside the parentheses to create a perfect square trinomial.
The result is a(x − h)² + k where h = −b/(2a) is the x-coordinate of the vertex and k = c − b²/(4a) is the y-coordinate. If a > 0, the vertex is a minimum; if a < 0, it is a maximum.
h = −b / (2a)
X-coordinate of the vertex.
k = c − b² / (4a)
Y-coordinate of the vertex.
Frequently asked questions
It converts a quadratic to vertex form, instantly revealing the vertex (minimum or maximum point), axis of symmetry, and whether the parabola opens up or down. It is also the algebraic basis for deriving the quadratic formula.
When a = 1, the vertex form simplifies to (x − h)² + k with no leading coefficient, making it easier to read. The formulas still apply: h = −b/2 and k = c − b²/4.
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