Completing the square

Completing the square rewrites a quadratic expression as a perfect-square expression plus or minus a constant. It can solve quadratic equations and derive the quadratic formula.

Perfect square trinomial: A trinomial that factors as the square of a binomial. The method constructs this form by adding the square of half the linear coefficient.
Binomial: A polynomial consisting of two terms. The expression being squared is a binomial such as x + b/2a.
Quadratic formula derivation: An algebraic derivation of the quadratic formula by completing the square in the general quadratic equation. The method gives a direct route from ax² + bx + c = 0 to the general root formula.
Factoring: Rewriting an algebraic expression as a product of simpler factors. Factoring solves quadratics directly when their roots yield accessible factors.
Quadratic equation: An equation in which the highest power of the unknown is two. Completing the square solves quadratics by isolating a squared expression.
Binomial square: The expansion (x + y)² = x² + 2xy + y², or its corresponding square with a minus sign. This identity shows which constant term completes a quadratic trinomial.
Conic section: A curve formed by intersecting a plane with a double cone, including circles, ellipses, parabolas, and hyperbolas. Completing squares in conic equations helps identify and classify their standard forms.
Quadratic formula method: Solving a quadratic equation by substituting its coefficients into the quadratic formula. Unlike completing the square, this method applies a ready-made formula without rearranging each equation.
Square root property: The principle that an equation of the form x² = k has solutions x = ±√k over the real numbers when k is nonnegative. After completing the square, taking square roots produces the solutions.
Polynomial: An expression formed from variables and coefficients using addition, subtraction, and nonnegative integer powers. A quadratic polynomial is the expression the method reorganizes.

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