Completing the Square Calculator
Convert any quadratic expression from standard form ax² + bx + c to vertex form a(x - h)² + k. This calculator finds the vertex, discriminant, and roots of the quadratic equation.
Completing the square is a technique that rewrites a quadratic expression into vertex form, which makes it easy to read off the vertex of the parabola, determine its direction of opening, and find its roots. The vertex form a(x - h)² + k reveals that the parabola has its vertex at (h, k), opens upward when a > 0, and opens downward when a < 0.
The process works by taking ax² + bx + c, factoring out a from the first two terms, and adding and subtracting the square of half the linear coefficient inside the parentheses. This yields a = coefficient, h = -b/(2a) for the x-coordinate of the vertex, and k = c - b²/(4a) for the y-coordinate.
Completing the square is also the derivation behind the quadratic formula. By applying the technique to the general equation ax² + bx + c = 0, you arrive at x = (-b ± sqrt(b² - 4ac)) / (2a). The discriminant b² - 4ac determines the nature of the roots: positive gives two distinct real roots, zero gives one repeated root, and negative gives two complex conjugate roots.