Quadratic Equation Solver

About the Quadratic Equation Solver

The quadratic equation solver finds the roots of any equation of the form ax^2 + bx + c = 0, where a, b, and c are real numbers and a is not zero. The tool handles the three cases that arise from the discriminant: two distinct real roots, one repeated root, or a pair of complex conjugate roots. When a equals zero, the equation degenerates to a linear one and is solved directly as bx + c = 0.

Quadratic equations were studied by ancient Babylonian mathematicians as early as 2000 BCE, according to tablet YBC 4663 (transliterated by Neugebauer, 1935). The closed-form solution, often called the quadratic formula, was given in geometric form by al-Khwarizmi in his 9th-century treatise Al-Jabr, from which the word "algebra" derives. The formula as we write it today requires negative numbers, which were not accepted in European mathematics until the 16th century.

A subtle numerical issue arises when the discriminant is small relative to b^2. In the standard formula x = (-b +/- sqrt(b^2 - 4ac)) / (2a), the subtraction -b + sqrt(b^2 - 4ac) suffers from catastrophic cancellation when b is positive and large. The tool detects this case and uses the alternative form q = -1/2 * (b + sign(b) * sqrt(b^2 - 4ac)), then computes the roots as x1 = q/a and x2 = c/q. This is the numerically stable variant recommended by Press et al. in Numerical Recipes §5.6.

All computation runs client-side using IEEE 754 double precision. Inputs may be any real numbers, and the tool reports complex roots explicitly with both real and imaginary parts rather than silently returning NaN.

How It Works

The quadratic formula gives the two roots directly:

x = (-b +/- sqrt(b^2 - 4ac)) / (2a)

The expression under the square root is called the discriminant, denoted Delta:

Delta = b^2 - 4ac

The sign of the discriminant determines the nature of the roots:

Delta > 0  : two distinct real roots
Delta = 0  : one repeated real root
Delta < 0  : two complex conjugate roots a +/- bi

When a = 0, the equation is no longer quadratic; it becomes linear (bx + c = 0) with the single solution x = -c / b. If both a and b are zero, the equation is either impossible (c != 0) or trivially true for all x (c = 0), and the tool reports each case explicitly.

For numerical stability, when |b| is large relative to |4ac|, the standard formula loses precision in the -b + sqrt(b^2 - 4ac) branch due to catastrophic cancellation. The alternative form, due to Press et al. (Numerical Recipes §5.6), computes q = -0.5 * (b + sign(b) * sqrt(b^2 - 4ac)) and then derives the roots as x1 = q / a and x2 = c / q. The product of the roots equals c / a (Vieta's formula), which provides a useful sanity check.

Worked Examples

Consider the default equation x^2 - 5x + 6 = 0 (a = 1, b = -5, c = 6). The discriminant is 25 - 24 = 1, which is positive, so there are two real roots: x = (5 +/- 1) / 2, giving x1 = 3 and x2 = 2. Both check by substitution: 3^2 - 5*3 + 6 = 9 - 15 + 6 = 0, and 2^2 - 5*2 + 6 = 4 - 10 + 6 = 0.

For a repeated root, take x^2 - 6x + 9 = 0 (a = 1, b = -6, c = 9). The discriminant is 36 - 36 = 0, so there is one repeated root: x = 6 / 2 = 3. This corresponds to the perfect square (x - 3)^2 = 0, which touches the x-axis at x = 3 without crossing it.

For complex roots, try x^2 + 1 = 0 (a = 1, b = 0, c = 1). The discriminant is 0 - 4 = -4, which is negative. The square root of -4 is 2i, so the roots are 0 +/- 1i, that is, +i and -i. The tool displays these as "0 +/- 1.000000i" rather than returning NaN. For a numerical edge case, x^2 + 1e8 x + 1 = 0 has discriminant 1e16 - 4, essentially equal to 1e16, and the naive formula would lose precision in the smaller root; the alternative form yields x1 = -1e-8 and x2 = -1e8 accurately.

When to Use This Tool

Use the quadratic equation solver when you need to:

• Find where a parabola crosses the x-axis in coordinate geometry.
• Compute the time at which a projectile reaches a given height under constant gravity (y = y0 + v0 t - 0.5 g t^2).
• Determine break-even points in economics when revenue and cost are modelled quadratically.
• Solve circuit resonance problems where the characteristic equation is quadratic.
• Find the roots of a polynomial factorisation during algebra homework.
• Verify a factorisation result by checking that the roots make the original polynomial evaluate to zero.
• Test edge cases such as a = 0, b = 0, or negative discriminants without writing code.

Limitations & Disclaimer

This solver applies the quadratic formula and its numerically stable variant to real coefficients. It does not handle symbolic algebra, complex coefficients, or higher-degree polynomials. The alternative-form stabilisation triggers only when b^2 is much larger than 4ac; in some intermediate cases precision may still be lost. Results are subject to IEEE 754 floating-point rounding. For exact or symbolic work, use a computer algebra system. See our disclaimer for full details.

Frequently Asked Questions

Last updated: September 9, 2026  ·  Author: HT99 Tools Editorial Team