The quadratic formula – step by step
Understand and solve quadratic equations with confidence
The quadratic equation ax² + bx + c = 0 is one of the most frequently asked topics in school mathematics. It arises whenever parabolas cross the x-axis, in optimization problems (minimum/maximum of a quadratic function), in physics (projectile motion, lever laws) and in engineering mathematics. The quadratic formula x = (−b ± √D) / (2a) always gives the result – provided the discriminant D is not negative.
The discriminant D = b² − 4ac is the decisive value. It sits under the square root and determines whether the equation has two (D > 0), one (D = 0) or no real solution (D < 0). If D < 0, the square root of a negative number would be required, which is impossible within the real numbers. The corresponding parabola f(x) = ax² + bx + c then does not cross the x-axis.
For simple equations with integer solutions, the factoring method is also efficient: for x² − 5x + 6 = 0, you look for two numbers with a sum of −5 and a product of +6. These are −3 and −2, so (x − 3)(x − 2) = 0 → x = 3 or x = 2. This method works without a calculator, but only when integer solutions exist. The quadratic formula always works – even for irrational solutions like x = (1 + √5) / 2 (the golden ratio).
In practice, quadratic equations appear in motion problems (solving s = v₀t + ½at² for t), area optimization (maximum area of a rectangle with a given perimeter) and circle geometry. Our calculator solves ax² + bx + c = 0 for any real coefficients a, b, c and shows the discriminant as well as all real solutions with six decimal places of precision.