Quadratic Equation Solver - Find Roots of ax²+bx+c=0

The discriminant D = b² − 4ac tells you the nature of the roots before you compute them, which is why it is the first thing you should calculate. If D is greater than zero the equation has two distinct real roots and the parabola crosses the horizontal axis twice. If D equals zero there is exactly one real root, repeated twice, and the parabola touches the axis at its turning point, which is the vertex. If D is less than zero there are no real roots at all, because the parabola opens away from the axis, and the two solutions form a complex conjugate pair of the form a + bi and a − bi. The magnitude of D also measures how far the roots sit from each other: the larger D is, the wider apart the two roots become. Working out the discriminant first is usually faster than substituting into the formula and discovering a negative square root halfway through.

This almost always means a is negative, or your plot has an incorrect scale, or you have transposed a coefficient. A parabola only crosses the horizontal axis if it opens upwards, meaning a must be positive, and it only touches the axis if a is positive and D equals zero. If a is negative the parabola opens downwards and stays entirely below the axis when D is less than zero, which is mathematically consistent even if your sketch appears to cross because the axes are not drawn to scale. Check first that you have not swapped b and c, since that error produces a plausible-looking graph with the wrong roots. Also confirm you did not omit a minus sign in front of a negative coefficient, a very common slip when copying from a textbook.

When a equals zero the equation ax² + bx + c = 0 is no longer quadratic because the squared term disappears. It reduces to the linear equation bx + c = 0, which has the single solution x = −c/b, provided b is also not zero. This is exactly the case where blindly applying the quadratic formula fails, because it divides by 2a and therefore divides by zero, producing an undefined or infinite result instead of the correct answer. If both a and b are zero, then the equation becomes the identity c = 0: if c is also zero every real number is a solution, and if c is not zero there are no solutions at all. Most solvers, including this one, detect these degenerate cases automatically and report which of the three situations applies rather than returning a division by zero.

You can use it for any equation that is genuinely quadratic, which means the coefficient a is not zero. What you cannot do is apply it directly to an equation that is not yet in standard form ax² + bx + c = 0. An equation such as x² + 6 = 5x must first be rearranged so that every term sits on one side, giving x² − 5x + 6 = 0. An equation containing fractions should be cleared of denominators first, for example x²/3 − x = 2 becomes x² − 3x = 6. An equation with a square root, a fraction containing x, or an exponent other than two is not quadratic at all and the formula does not apply, so substitution or a different method is needed. The formula also works numerically for any real coefficients, so you can enter decimals or large numbers directly.

All three produce the same roots, and the difference is how much insight they give you. The quadratic formula is a mechanical procedure that works for every quadratic including ones that cannot be factored with whole numbers, which makes it the safest general approach and the one to fall back on in an exam. Completing the square rewrites the equation into (x − h)² = k, which reveals the vertex directly and shows you why the discriminant controls whether real roots exist, though it involves more algebra and introduces fractions easily. Factoring writes the equation as (x − x₁)(x − x₂) = 0, which gives roots fastest when the numbers are integers and teaches the structure behind the whole method, but it fails outright when the roots are irrational or complex. Completing the square is the conceptual bridge between the other two, which is why it is worth working through at least once by hand.

You get complex roots precisely when the discriminant is negative, meaning b² − 4ac is less than zero. That is not an error in the tool or a sign you entered incorrectly; it is the mathematically correct answer for an equation whose parabola never intersects the horizontal axis. The roots take the form a + bi and a − bi, where b is not zero and i is the imaginary unit. In most school contexts this means the equation has no real solution, and that is the result you should report. Complex roots appear legitimately in fields such as electrical engineering, where they describe oscillatory behaviour and circuit impedance, and in quantum mechanics, where real-valued solutions do not exist. If you were expecting real numbers, re-check b² and 4ac separately, since a small arithmetic slip often flips the sign of the discriminant.