How to Use a Quadratic Equation Solver Step by Step

To solve a quadratic with an online solver, enter the three coefficients a, b and c from ax² + bx + c = 0, and read back the roots, the discriminant, the vertex and the axis of symmetry — each with the working shown. The ByteTools Quadratic Equation Solver applies the quadratic formula for you and handles real or complex roots automatically. Here is the full walkthrough plus what each result means.

What the solver produces

From any ax² + bx + c = 0, the solver computes the discriminant b² − 4ac to classify the roots, then returns the roots themselves, the vertex of the parabola, and its axis of symmetry — showing every substitution rather than just an answer. When the discriminant is negative it does not give up; it returns the two complex conjugate roots in exact p ± qi form. Everything runs locally in your browser with JavaScript, so nothing is uploaded, results are instant, and you can copy the roots or the full solution with one click.

Step by step

1. Enter a, the x² coefficient. This is the leading term. If it is zero the equation is not quadratic — see the note below.
2. Enter b and c. b is the x coefficient and c is the constant. Include their signs; a minus sign matters.
3. Read the discriminant. The solver shows b² − 4ac and tells you whether the roots are two real, one repeated real, or two complex.
4. Review roots, vertex and axis. The roots come from the quadratic formula, the vertex from x = −b ÷ (2a), and the axis of symmetry passes vertically through it.
5. Copy the solution. Grab just the roots or the full worked steps for homework or a report.

A worked example

Take x² − 5x + 6 = 0, so a = 1, b = −5, c = 6. The discriminant is (−5)² − 4(1)(6) = 25 − 24 = 1, which is positive, so there are two distinct real roots. The formula gives x = (5 ± √1) ÷ 2, yielding x = 3 and x = 2. The vertex sits at x = 5 ÷ 2 = 2.5, and the axis of symmetry is the vertical line x = 2.5.

Discriminant b² − 4ac	Root type	Example
Positive	Two distinct real roots	x² − 5x + 6 → 2, 3
Zero	One repeated real root	x² − 4x + 4 → 2
Negative	Two complex conjugates	x² + x + 1 → complex

Why it runs in your browser

Because the maths happens on your device, your coefficients never leave the page — convenient for graded work you would rather keep private, and it means the solver still works with no connection. The instant recalculation as you change a value also makes it a fast way to explore how coefficients reshape a parabola.

Try the Quadratic Equation Solver — free and 100% in your browser.

FAQ

What do I enter if a term is missing?

Use zero for the missing coefficient. For x² − 9 = 0, enter a = 1, b = 0, c = −9. The solver treats the absent x term as b = 0 and returns x = 3 and x = −3.

What happens if I set a to zero?

The equation becomes linear, not quadratic, so the quadratic formula does not apply. The solver detects this and solves bx + c = 0 instead, or flags the case when b is also zero and there is nothing to solve.

Does it really solve equations with no real answer?

Yes. When the discriminant is negative, instead of reporting no solution the solver gives the two complex conjugate roots in p ± qi form, using the square root of the absolute discriminant.

Can I get the vertex without solving for roots?

The vertex is computed alongside the roots automatically. It comes from x = −b ÷ (2a) with the y-value found by substituting back, so you get it in the same result whether or not the roots are real.

Related free tools

• Scientific Calculator — powers, roots and functions
• Pythagorean Theorem Calculator — right-triangle sides
• Percentage Calculator — quick percentage maths
• GCD & LCM Calculator — factors and multiples

Built by ByteVancer

ByteTools is a free product of ByteVancer, a software and web development studio building web apps, SaaS, and custom software. If you need a custom calculator, teaching tool, or a full web app, explore what ByteVancer can build for you.