Solve quadratic equations ax² + bx + c = 0 using the quadratic formula. Find real and complex roots with full working.
Coefficient of x²
Coefficient of x
Constant term
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x = (−b ± √(b² − 4ac)) / 2aaCoefficient of x² (must not be 0)bCoefficient of xcConstant termΔDiscriminant = b² − 4acThe quadratic formula solves any equation of the form ax² + bx + c = 0. The discriminant (b² − 4ac) determines the nature of the roots: positive → two real roots, zero → one repeated root, negative → two complex conjugate roots.
The quadratic equation is one of the most fundamental concepts in algebra, appearing in physics (projectile motion), engineering (structural analysis), economics (profit maximization), and computer graphics (curve rendering). Mastering quadratic equations opens the door to understanding parabolas, optimization problems, and polynomial functions.
Enter the three coefficients a, b, and c of your quadratic equation in the form ax² + bx + c = 0. The solver instantly computes the discriminant, both roots (real or complex), and the vertex of the parabola, with a complete step-by-step solution.
This solver handles all cases - two real roots, one repeated root, and complex conjugate roots - making it a complete solution for any quadratic equation. The vertex calculation is a bonus feature useful for graphing and optimization problems.
The most common mistake is entering the wrong sign for b or c. Remember that if your equation is x² − 5x + 6 = 0, then a = 1, b = −5, c = 6. Also, coefficient a must never be zero; if it is, the equation is linear, not quadratic.
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