We use cookies.This website uses essential cookies to operate core features. With your consent, we also use analytics cookies to understand traffic and improve the service. For more details, see our .
We use cookies.This website uses essential cookies to operate core features. With your consent, we also use analytics cookies to understand traffic and improve the service. For more details, see our .
Was this tool helpful to use?
Your feedback helps us make it better
Enter coefficients a, b, and c to instantly calculate the real or complex roots of a quadratic equation.
Enter coefficients a, b, and c to see the roots.
Overview
Understand what the tool solves, how it works, and the boundaries of its data.
This calculator solves equations written as ax² + bx + c = 0. Enter the numeric coefficients a, b, and c in their matching fields; the tool reports the solution type and roots. For a genuine quadratic, a must be nonzero. If your equation is written differently, first expand parentheses, collect like terms, and move every term to one side so the right side is zero. Keep each coefficient’s sign: in −6x² + 3x − 2 = 0, for example, a = −6, b = 3, and c = −2.
The quadratic formula is x = (−b ± √Δ)/(2a), where Δ = b² − 4ac is the discriminant. It is the quantity under the square root, so its sign identifies the type of roots: a positive Δ gives two distinct real roots, zero gives one repeated real root, and a negative Δ gives a conjugate pair of complex roots. This relationship and formula are summarized in OpenStax College Algebra, Section 2.5.
For a quadratic, results include the discriminant, roots, a substitution based solution outline, and a plotted view of y = ax² + bx + c. When Δ is negative, the roots appear in the form u ± vi, where u is the real part and v is the magnitude of the imaginary part. The graph is a visual aid for the real valued parabola; it does not plot complex roots as x-intercepts. Results are numerical and formatted to a limited number of significant digits, rather than returned as symbolic expressions such as a simplified radical.
A zero value of a makes the expression linear or constant, and the calculator handles that case separately instead of applying the quadratic formula. With a = 0 and b ≠ 0, it solves bx + c = 0. With a = b = 0, c = 0 makes the equation true for every x, while nonzero c means there is no solution.
Guide
Follow the workflow and verify inputs and outputs with practical examples.
Rewrite the problem as ax² + bx + c = 0. Expand brackets and move terms across the equals sign before identifying coefficients.
Enter a, b, and c in order. Include negative signs and enter zero for a missing term; for example, x² − 9 = 0 uses a = 1, b = 0, c = −9. Decimal values are accepted. All three fields need a numeric value.
Read the root or roots, then inspect Δ and the displayed working. A quadratic also has a graph of y = ax² + bx + c. Use the copy controls if you want to paste the result elsewhere, or reset the fields to start another calculation.
Check a real root by substituting it into the original equation. For rounded decimal roots, the left side may be close to zero rather than exactly zero. For a complex pair, check the real and imaginary parts algebraically.
Example: For 2x² − 4x − 6 = 0, enter a = 2, b = −4, c = −6. Then Δ = (−4)² − 4(2)(−6) = 64, and x = (4 ± 8)/4, giving x = 3 and x = −1. Substitution confirms both: 2(3²) − 4(3) − 6 = 0, and 2(−1²) − 4(−1) − 6 = 0.
Use cases
See how the tool fits into real work and everyday tasks.
A student who has already rearranged an equation can enter its coefficients and compare the discriminant, roots, and formula substitution with their own work. A mismatch often points to a missed sign or a coefficient copied into the wrong field.
A tutor or study group can hold a and b steady while changing c, then observe how the discriminant and graph change. Discuss whether the parabola crosses, touches, or misses the x-axis; the plotted curve is a numerical visual aid.
An instructor or editor preparing a lesson can verify a concrete equation and copy its computed roots. If a task requires factoring, completing the square, exact radical notation, or a proof, use the calculator’s result as a check rather than as a substitute for that requested method.
Q&A
Find concise answers to common questions and confusing cases.
No. Enter the three numeric coefficients separately. Rearrange the equation into ax² + bx + c = 0 first; this calculator does not parse arbitrary text or perform symbolic simplification.
A negative Δ has no square root in the real number system. Including the imaginary unit i gives two conjugate solutions, displayed as u ± vi. They are solutions over the complex numbers, even though the real graph has no x-intercept.
The equation is no longer quadratic. If b is nonzero, the calculator solves bx + c = 0; if b is also zero, it reports an identity when c is zero and no solution otherwise.
The roots and intermediate values are calculated numerically and formatted to limited precision. Check with unrounded values if available, or substitute the displayed decimals into the original equation and expect a small residual from rounding.
Notes
Review scope, result limitations, and important precautions before use.
This tool is for equations with one variable and numeric coefficients. It does not solve cubic or higher degree equations, accept variables as coefficients, interpret word problems, or return exact symbolic forms. Values are displayed with finite precision, and the graph is sampled and rounded for visualization; neither should be treated as an exact proof. For graded work, follow the requested solution method and show the algebraic steps your assignment requires.
Related
Discover related tools, collections, and available API capabilities.