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Solve ax² + bx + c = 0 (a ≠ 0).
For ax² + bx + c = 0 the discriminant is D = b² − 4ac and the roots are (−b ± √D) / 2a. Example check: x² − 5x + 6 = 0 gives D = 25 − 24 = 1, so the roots are (5 ± 1) / 2 = 3 and 2. A negative discriminant gives complex roots, labelled as complex — not “no answer”. If a = 0 the equation is linear, not quadratic, and the tool refuses it.
Try an example: x² − 5x + 6 = 0 (roots 3 and 2)
The Quadratic Equation Solver is a free online tool that solves any quadratic equation of the form ax squared plus bx plus c equals zero, where a is not zero. You enter the three coefficients a, b, and c, and the tool works out the discriminant, finds the roots, and shows the vertex and the axis of symmetry, all with step-by-step working you can follow. It runs entirely in your browser, needs no sign-up, and gives you the answer in seconds, whether the roots are real numbers or complex numbers.
A quadratic equation is an equation where the highest power of the unknown is two. It appears everywhere in maths and science, from the path of a thrown ball to the shape of a satellite dish, from profit curves in business to the design of bridges and lenses. Every quadratic has the same standard form, ax squared plus bx plus c equals zero, and the quadratic formula solves all of them in one stroke. Learning to use the formula is a key step in algebra, and this tool lets you practise it with instant feedback.
The heart of the method is the discriminant, written as D, and defined by the formula D = b squared minus 4ac. The discriminant tells you, before you finish the calculation, what kind of roots the equation has. If D is greater than zero, there are two different real roots. If D equals zero, there is exactly one real root, repeated twice. If D is less than zero, there are two complex roots. The tool calculates D first and shows it clearly, so you always know which case you are in.
Once the discriminant is known, the roots come from the quadratic formula: x equals minus b plus or minus the square root of D, all divided by 2a. The tool applies this formula with your numbers, showing the substitution, the value of the square root, and the two separate calculations for the plus and the minus cases. Nothing is hidden, so you can copy the working into your homework or check it against your own attempt line by line.
Here is the worked example the tool uses. Take the equation x squared minus 5x plus 6 equals zero, so a is 1, b is minus 5, and c is 6. The discriminant is D = b squared minus 4ac, which is 25 minus 24, which equals 1. The roots are then x = (5 plus or minus 1) divided by 2, which gives x = 3 and x = 2. You can check this by substituting back: 3 squared minus 15 plus 6 is 0, and 2 squared minus 10 plus 6 is also 0. The tool shows every one of these steps.
When the discriminant is negative, many simple solvers give up and say there is no answer, but that is not correct, and this tool does not do that. A negative discriminant means the roots are complex numbers, and the tool shows them labelled as complex, with their real and imaginary parts clearly marked. For example, an equation with D equal to minus 4 has roots involving the square root of minus 4, which is 2i, and the tool writes those roots out in full. Complex roots are real mathematical answers, and they matter in engineering, physics, and advanced maths, so the tool treats them properly instead of hiding them.
The tool is also honest about the one input it will not accept. It refuses a = 0, because if a is zero, the x squared term disappears and the equation is linear, not quadratic. Solving a linear equation with a quadratic solver would give a misleading or meaningless result, so the tool explains the problem and asks you to enter a non-zero value for a. That clear refusal is a feature, not a limitation, because it stops a common mistake at the source and reminds you what makes an equation quadratic in the first place.
Beyond the roots, the tool also finds the vertex and the axis of symmetry of the parabola, which is the curved graph of the quadratic. The vertex is the turning point of the curve, its highest or lowest point, and its x-coordinate is minus b divided by 2a. The axis of symmetry is the vertical line through the vertex, x equals minus b divided by 2a, and the parabola is a mirror image on either side of it. Knowing the vertex tells you the maximum or minimum value of the function, which is exactly what you need when a quadratic models height, profit, or cost.
Using the tool is simple. Type the value of a into the first box, b into the second, and c into the third, then press solve. The tool shows the discriminant, the nature of the roots, the roots themselves, the vertex, and the axis of symmetry, together with the step-by-step working. Try the worked example first, with a equal to 1, b equal to minus 5, and c equal to 6, and confirm that you get D equal to 1 and roots 3 and 2. Then try your own equations, including one with a negative discriminant, and watch how the tool handles the complex roots.
Students get the most from the tool by working alongside it rather than copying from it. Solve the equation on paper first, using the discriminant and the quadratic formula, then enter the same coefficients and compare each step with the tool. If your discriminant is wrong, everything after it will be wrong too, so that is the first place to check. If your roots differ, look at the plus and minus steps separately, because sign errors in minus b are the most common mistake. Teachers can use the tool to build worked examples, and the vertex and axis of symmetry make it useful for graphing lessons as well.
The quadratic formula connects to many other ideas in algebra. Factoring is the same problem in reverse: the roots 3 and 2 from the worked example mean the equation factors as x minus 3 times x minus 2. Completing the square is the method from which the formula itself is derived. The discriminant is also the key to sketching a parabola, because it tells you whether the curve crosses the x-axis twice, touches it once, or never meets it at all. Seeing the roots, the vertex, and the discriminant together in one place helps all of these ideas click into place.
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To keep working with numbers, try the Fraction Calculator for fraction arithmetic with simplification, the Statistics Calculator for dataset statistics, and the Unit Converter for converting between units. All three are free, run in your browser, and sit naturally alongside this solver in any maths toolkit.
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