Solve any quadratic equation in the form ax² + bx + c = 0 using the quadratic formula.
Google AdSense Banner
This area will contain advertisements after approval.
Google AdSense Banner
This area will contain advertisements after approval.
The quadratic formula solves ax² + bx + c = 0 for x: x = (−b ± √(b² − 4ac)) / (2a). The term b² − 4ac is called the discriminant, and its sign determines whether the equation has two real roots, one repeated real root, or two complex roots.
A positive discriminant means two distinct real roots. A discriminant of exactly zero means one repeated real root (the parabola touches the x-axis at a single point). A negative discriminant means the roots are complex numbers, meaning the parabola never crosses the x-axis.
For x² − 3x + 2 = 0: a = 1, b = −3, c = 2. The discriminant is b² − 4ac = (−3)² − 4(1)(2) = 9 − 8 = 1, which is positive, so there are two real roots. Applying the formula: x = (3 ± √1) / 2, giving x₁ = (3 + 1)/2 = 2 and x₂ = (3 − 1)/2 = 1.
Mixing up the signs of b and c when reading them out of the original equation, for x² − 3x + 2 = 0, b is −3 and c is +2, not 3 and −2, a sign slip here throws off both roots.
Forgetting the ± in the quadratic formula and only computing one root, every quadratic with a positive discriminant has two distinct solutions, not one.
Assuming a negative discriminant means there's no answer at all, rather than that the roots are complex numbers; the equation still has solutions, they're just not real numbers.
If a equals zero, the equation isn't quadratic anymore, it becomes linear. This calculator requires a nonzero value for a.
A complex root includes an imaginary component (shown with 'i'), meaning the equation's graph doesn't intersect the x-axis at any real number.
Google AdSense Banner
This area will contain advertisements after approval.