A fast, private quadratic equation solver
This quadratic equation solver provides instant, comprehensive solutions for any second-degree polynomial. Whether you are a student checking your algebra homework, a teacher generating examples, or an engineer solving for variables, this tool gives you the exact answers and the mathematical steps required to reach them.
Because the tool runs entirely in your browser, your equations are never sent to a server. You get immediate results as you type, with complete privacy and no waiting.
How to solve ax2+bx+c=0
We designed the input to be as flexible as possible. You don't need to manually separate your equation into coefficients if you don't want to.
You can simply type your full equation into the main text box. For example, entering 18x^2 - 51x + 15 = 0 will automatically parse the string, identify the variable (whether you use $x$, $y$, $t$, or another letter), and extract the coefficients. The tool handles negative signs and standardizes the format for you.
Alternatively, if you already know your coefficients, you can enter them directly into the $a$, $b$, and $c$ input fields. Updating any field instantly recalculates the entire problem, making it easy to see how changing a single variable affects the parabola and its roots.
Understanding your results
Once you enter a valid equation, the calculator breaks down the problem into several key mathematical components:
Roots of quadratic equation: The tool calculates the exact points where the parabola crosses the x-axis. Depending on the equation, you will see two real roots, one repeated real root, or two complex roots (involving $i$, the imaginary unit).
Discriminant calculator: The discriminant ($\Delta = b^2 - 4ac$) is the part of the quadratic formula under the square root. The tool calculates this value first, as it dictates the nature of the roots. A positive discriminant means two real roots, a zero means one real root, and a negative discriminant means complex roots.
Factor quadratic expressions: Whenever possible, the tool will display the factored form of your equation (e.g., $(2x - 5)(x + 3) = 0$). This is particularly useful for algebra students who need to factor by grouping or use the AC method. If the equation's roots are irrational or complex, the tool will simply note that it is not cleanly factorable.
Vertex and Axis of Symmetry: Every quadratic equation graphs as a parabola. The calculator finds the vertex—the absolute minimum or maximum point of that curve. It also calculates the axis of symmetry, which is the vertical line that splits the parabola into two mirrored halves.
Step-by-step quadratic formula calculator
Simply knowing the final answer isn't always enough, especially if you are learning algebra or trying to debug a larger physics problem. To help you understand the process, the tool generates a step-by-step breakdown of how the quadratic formula is applied to your specific numbers.
The steps section shows:
- The extraction of the $a$, $b$, and $c$ coefficients.
- The substitution of those numbers into the discriminant formula ($\Delta = b^2 - 4ac$) and the resulting value.
- The application of the full quadratic formula, $x = (-b \pm \sqrt{\Delta}) / 2a$, showing the exact arithmetic used to find the roots.
- The calculation for the axis of symmetry, $x = -b / 2a$, which provides the x-coordinate of the vertex.
Related math tools
If you are working through complex math assignments, you might find our other local, browser-based calculators helpful. For general purpose calculations, use the Scientific Calculator. If your quadratic coefficients involve fractions that need simplifying, the Fraction Calculator can help. Finally, for finding common denominators or simplifying ratios before plugging them into the quadratic formula, try the GCD & LCM Calculator.