Quadratic Equation Solver

A quadratic equation has the general form ax² + bx + c = 0, where a, b and c are numbers, called coefficients, and a cannot be zero — if it were, the x² term would disappear and the equation would just be a straight line, linear, not a curve, quadratic. Quadratics describe parabolas: the path of a thrown ball under gravity, the shape of a satellite dish, the profit curve of a business as price changes, and countless other real-world relationships where a quantity rises, peaks or dips, and falls. "Solving" a quadratic equation means finding the value or values of x that make the equation true, where the parabola crosses the x-axis. The standard tool for this is the quadratic formula: x = (−b ± √(b² − 4ac)) / (2a). The "±" symbol means there are potentially two answers: one using +√(...) and one using −√(...). The expression under the square root, b² − 4ac, is called the discriminant, and its sign alone tells you what kind of answer to expect before you even finish the calculation. If the discriminant is positive, its square root is a normal real number, so the ± gives two distinct real roots — the parabola crosses the x-axis at two separate points. If the discriminant is exactly zero, the ± collapses to a single value, since adding or subtracting zero makes no difference, giving one repeated real root — the parabola just touches the x-axis at a single point, its vertex. If the discriminant is negative, you'd be taking the square root of a negative number, which isn't a real number, so the equation has two complex roots instead, and the parabola never touches the x-axis at all. Worked example with two real roots: x² − 5x + 6 = 0, where a=1, b=−5, c=6. Discriminant = (−5)² − 4(1)(6) = 25 − 24 = 1, which is positive, so there are two real roots: x = (5 ± 1)/2, giving x = 3 and x = 2. You can verify this by substituting back in: 3² − 5(3) + 6 = 9 − 15 + 6 = 0, correct. Worked example with complex roots: x² + 2x + 5 = 0, where a=1, b=2, c=5. Discriminant = 2² − 4(1)(5) = 4 − 20 = −16, which is negative. The real part of each root is −b/(2a) = −2/2 = −1, and the imaginary part is √16/2 = 4/2 = 2, so the two roots are −1 + 2i and −1 − 2i, a "complex conjugate pair", where the imaginary parts have opposite signs but everything else matches. A common misconception is thinking a negative discriminant means the equation has "no solution" at all. It does have solutions — they're just not real numbers, they're complex numbers, which are essential in electrical engineering (AC circuits), control systems, and quantum mechanics. Another frequent error is forgetting the ± entirely and reporting only one root when the discriminant is positive, missing half of the valid answers.