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For the equation ax² + bx + c = 0
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Solve quadratic equations ax²+bx+c=0 instantly. Find real and complex roots with step-by-step solutions, discriminant, and vertex. Free browser-based calculator.
For the equation ax² + bx + c = 0
Enter coefficients and solve
Results will appear here with step-by-step breakdownFill in a, b, and c for your equation ax²+bx+c=0. Coefficient a must not be zero.
Hit the Solve button or press Enter. Results appear instantly with all working shown.
View real or complex roots, discriminant value, vertex, and the parabola graph.
A quadratic equation is any polynomial of the form ax²+bx+c=0 where a≠0. This solver uses the quadratic formula x = (−b ± √(b²−4ac)) / 2a to find exact roots — real or complex — instantly.
A quadratic equation is a second-degree polynomial equation of the form ax²+bx+c=0, where a, b, and c are constants and a≠0. It always has exactly two roots (counting multiplicity), which may be real or complex numbers.
The discriminant Δ = b²−4ac determines the nature of the roots: if Δ > 0 there are two distinct real roots; if Δ = 0 there is one repeated real root; if Δ < 0 the roots are complex conjugates with no real solutions.
Yes. This solver accepts any numeric input including decimals and negative numbers for all three coefficients a, b, and c. Just type them directly into the input fields.
When the discriminant is negative, the square root of a negative number appears in the quadratic formula. The result is a pair of complex conjugate roots of the form p ± qi, where i = √(−1). These roots are real-world relevant in AC circuits, oscillations, and signal processing.
The vertex is the turning point of the parabola y = ax²+bx+c. Its x-coordinate is −b/(2a) and the y-coordinate is c − b²/(4a). If a > 0 the vertex is a minimum; if a < 0 it is a maximum.
Completely free. All calculations run in your browser using JavaScript — no data is sent to any server, and no sign-up is required.
A quadratic equation solver is a tool that finds the values of x that satisfy any equation of the form ax² + bx + c = 0. Instead of working through the quadratic formula by hand — which is error-prone and time-consuming — this calculator handles everything instantly: real roots, complex roots, the discriminant, the vertex, and even a visual parabola graph.
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The quadratic formula is one of the most famous results in all of mathematics:
x = (−b ± √(b² − 4ac)) / 2a
Given any quadratic equation ax²+bx+c=0 with a≠0, this formula always produces the correct roots. The ± symbol means there are generally two roots — one computed with addition and one with subtraction. If those roots turn out to be identical (when the discriminant equals zero), the equation has a single repeated root.
Before you even compute the roots, the discriminant tells you what kind of solution to expect:
This diagnostic step is crucial in applied mathematics and engineering, where knowing the nature of the solution before computing it helps guide interpretation.
When Δ ≥ 0, the roots are real numbers that can be plotted on a number line and found as x-intercepts of the parabola y = ax²+bx+c. When Δ < 0, the formula requires the square root of a negative number. Rather than being undefined, this gives a complex number result of the form:
x = (−b ± i·√|Δ|) / 2a
Complex roots always appear in conjugate pairs: if p + qi is a root, then p − qi is the other. They are essential in electrical engineering (impedance), control theory (stability analysis), and quantum mechanics.
Every quadratic function y = ax²+bx+c traces a parabola. The vertex is its highest or lowest point, depending on the sign of a:
When a > 0, the parabola opens upward and the vertex is a minimum. When a < 0, it opens downward and the vertex is a maximum. This is used constantly in optimization problems — finding minimum cost, maximum area, or peak profit.
This solver shows all working so you can learn alongside the result:
Seeing each step makes the quadratic formula far easier to remember and apply independently.
Quadratic equations appear throughout mathematics and applied science:
Manual calculation is straightforward for simple coefficients but quickly becomes error-prone with fractions, large numbers, or negative values. This browser-based solver eliminates arithmetic mistakes, handles complex roots automatically, displays all intermediate steps, and renders the parabola graph — all without any installation or sign-up. It runs entirely in your browser with no data sent to external servers.