Which formula is used to solve any quadratic equation?

• A. Completing the Square
• B. Descartes' Rule of Signs
• C. Quadratic Formula
• D. Rational Zero Theorem

Answer: (C)

To solve any quadratic equation in the form ax^2 + bx + c = 0 with a ≠ 0, you use the Quadratic Formula: x = (-b ± sqrt(b^2 - 4ac)) / (2a). This closed-form expression works for all coefficients, giving two roots (counting multiplicity) that may be real or complex depending on the discriminant b^2 - 4ac. The discriminant tells you the nature of the roots: positive means two real roots, zero means a double root, and negative means two complex conjugates. This formula is universal because you can apply it directly to any quadratic without additional manipulation, and it’s actually derived by completing the square. Other methods exist, like completing the square (a technique that leads to the formula), Descartes' Rule of Signs (which counts potential real roots but doesn’t solve), and the Rational Zeros Theorem (which helps identify possible rational roots but doesn’t always yield the full solution). In short, the Quadratic Formula is the standard tool for solving any quadratic equation.