Completing the Square Calculator — Show Steps
Vertex Form
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Vertex (h, k)
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Roots
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Steps
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How to Complete the Square
Completing the square transforms a quadratic expression ax^2 + bx + c into vertex form a(x - h)^2 + k, where (h, k) is the vertex of the parabola. The vertex form makes it easy to identify the minimum or maximum value and the axis of symmetry.
The process: factor out a from the first two terms, take half the coefficient of x, square it, add and subtract inside the parentheses, then simplify. The vertex is at h = -b/(2a) and k = c - b^2/(4a).
Completing the square is used to derive the quadratic formula, solve quadratic equations, convert conic section equations to standard form, and find maximum/minimum values in optimization problems. It is a fundamental technique in algebra.
Frequently Asked Questions
What is vertex form?
Vertex form is y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. If a > 0, the vertex is the minimum point. If a < 0, it is the maximum point.
Why is completing the square useful?
It reveals the vertex of a parabola, makes it easy to solve quadratic equations, and is used to derive the quadratic formula. It also helps in converting circle and ellipse equations to standard form.
How does this relate to the quadratic formula?
The quadratic formula x = (-b +/- sqrt(b^2-4ac))/(2a) is derived by completing the square on the general form ax^2 + bx + c = 0. The discriminant b^2-4ac determines the nature of the roots.