Quadratic Formula Calculator

Any quadratic equation can be expressed in standard form as ax² + bx + c = 0, where 'a' does not equal zero. The primary mechanism for solving this equation is the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a). The term underneath the square root, known as the discriminant (Δ = b² - 4ac), dictates the nature of the roots. If the discriminant is positive, the equation yields two distinct real roots. If it equals zero, there is exactly one repeated real root. If it is negative, the equation produces a pair of conjugate complex roots. Additionally, the calculator determines the vertex form coordinates (h, k) using the formulas h = -b / (2a) and k = c - b² / (4a).

Let us solve the quadratic equation 2x² - 8x + 6 = 0. Here, our coefficients are a = 2, b = -8, and c = 6. First, we compute the discriminant: Δ = (-8)² - 4(2)(6) = 64 - 48 = 16. Since the discriminant is positive (16 > 0), we expect two distinct real roots. Next, we apply the quadratic formula: x = (-(-8) ± √16) / (2 * 2), which simplifies to (8 ± 4) / 4. For our first root, x₁ = (8 - 4) / 4 = 4 / 4 = 1. For our second root, x₂ = (8 + 4) / 4 = 12 / 4 = 3. Finally, to find the vertex coordinates, we calculate h = -(-8) / (2 * 2) = 8 / 4 = 2, and k = 6 - (-8)² / (4 * 2) = 6 - (64 / 8) = 6 - 8 = -2, giving a vertex of (2, -2).

Always ensure your quadratic equation is arranged in standard form (ax² + bx + c = 0) before identifying your coefficients. Pay close attention to negative signs, especially when squaring negative values for the 'b' term or multiplying through with 'a' and 'c'. If your calculation yields a negative discriminant and you require only real-world solutions, double-check your input values or enable complex number mode if your application permits imaginary roots.