What is a quadratic formula calculator?
A quadratic formula calculator finds the values of x that satisfy an equation of the form ax² + bx + c = 0. A true quadratic has a nonzero coefficient a. Its graph is a parabola, and its roots are the values that make the expression zero. This quadratic equation solver reports two distinct real roots, one repeated real root, or two complex conjugate roots.
The three inputs are coefficients, not complete equations. Rewrite x² − 5x = −6 as x² − 5x + 6 = 0 before entering a = 1, b = −5 and c = 6. If the problem has only a first power of x, the Linear Equation Calculator offers a focused alternative.
How to enter quadratic coefficients
- Move all terms to one side and combine like terms.
- Enter a, including its sign; enter 1 for an unwritten coefficient of x².
- Enter b and c, using 0 where a term is absent.
- Select Calculate and inspect both the root classification and the calculation steps.
Decimal and scientific-notation values are accepted within the stated limits. Expressions such as 1/3 are not accepted, because silently interpreting an expression as a decimal would obscure its precision. Use the Fraction Calculator when preparing rational arithmetic, and remember that a rounded decimal coefficient defines a slightly different equation from an exact fraction.
Quadratic formula, discriminant and numerical method
The discriminant D determines the type of roots. Positive D produces two distinct real roots; zero D gives a repeated real root; negative D gives two complex roots rather than real x-axis intersections. The tool first converts entered decimal coefficients to equivalent integer coefficients and removes a common factor. This keeps the discriminant sign exact for the input values instead of deciding the root type from a rounded decimal.
Real-root evaluation uses an algebraically equivalent stable form when D is positive: calculate q = −½(b + sign*(b)√D), with sign*(0) = +1, then obtain q/a and c/q. This reduces subtraction cancellation for roots with very different sizes. The displayed steps identify this method; no alternative formula is hidden from the explanation.
Understanding real and complex answers
A repeated root appears once but has multiplicity two. A pair such as 0 ± 1i represents two separate values: i and −i. Here i² = −1. Complex roots are valid mathematical solutions, but a physical problem may require a real, positive or otherwise restricted value.
The tool substitutes unrounded roots back into the equivalent integer-coefficient equation and shows residuals. A small nonzero residual can result from floating-point evaluation; it is not another root. Printed roots use up to 12 significant digits. To view the shape of the original polynomial, enter it into the Graphing Calculator; a graph is useful context but does not replace the exact discriminant classification.
Worked quadratic formula examples
For x² − 5x + 6 = 0, the discriminant is 25 − 24 = 1. The formula produces (5 ± 1)/2, so the roots are 2 and 3. Substitution confirms 4 − 10 + 6 = 0 and 9 − 15 + 6 = 0.
For x² + 1 = 0, D = −4 and the roots are ±i. For x² − 2x + 1 = 0, D = 0 and the repeated root is 1. Comparing these three examples shows why the discriminant is more informative than a single displayed number.
Coefficient precision and nonquadratic cases
Each nonzero coefficient must have magnitude between 10⁻¹² and 10¹², with no more than 12 significant digits. This is a bounded educational numerical solver, not an arbitrary-precision symbolic algebra system. Extremely close roots can look identical after display rounding even when the exact discriminant is positive.
If a = 0, the tool does not divide by 2a. It solves bx + c = 0 when b is nonzero. If a and b are both zero, c = 0 gives infinitely many solutions, while nonzero c gives no solution. Always check the original problem's domain: a negative time or length can satisfy an equation without being an acceptable practical answer.
Quadratic Formula Calculator FAQs
Sources and methodology
The references below explain the mathematical definitions and methods. This independently implemented tool is not affiliated with these publishers. Its displayed steps identify the actual method and precision used.