What is Quadratic Formula Solver?

Solve any quadratic equation using the quadratic formula. Enter coefficients a, b, and c to get both roots, the discriminant, vertex coordinates, and a visual graph of the parabola. Handles real and complex roots.

Punch in the three coefficients and you get both roots, the discriminant, the vertex coordinates, the axis of symmetry, and a parabola plot drawn in real time. Negative discriminants give complex roots in a + bi form. The step-by-step solution shows the formula substitution so it works for homework checks, not just answers.

How to use

1. Enter the coefficient values for a, b, and c in the equation ax² + bx + c = 0.
2. View both roots (real or complex), the discriminant, and the vertex of the parabola.
3. Copy the step-by-step solution or download the results for your homework or project.

When to use

• Checking algebra homework where you already worked out roots by hand.
• Finding where a projectile, profit curve, or any parabolic model crosses zero.
• Visualising how changing a coefficient flips or stretches the parabola.

Result

Solving 2x² - 4x - 6 = 0: enter a=2, b=-4, c=-6. The discriminant is 64 (positive, two real roots). Solutions: x = 3 and x = -1. Vertex at (1, -8).

FAQ

What does the discriminant actually tell me?

It's b² minus 4ac, the part under the square root. Positive means two real roots, zero means one repeated root, negative means two complex roots. The sign alone tells you what kind of solution you'll get before solving.

Why does the tool reject a = 0?

If a is zero the equation collapses to bx + c = 0, which is linear, not quadratic. There's no parabola and the formula divides by zero. Use a linear equation solver instead, or set b and c yourself.

How are complex roots written?

When the discriminant is negative, the roots come out as a pair like -1 + 2i and -1 - 2i. The real part is -b/(2a), the imaginary part is √|discriminant|/(2a). They always come as a conjugate pair for real coefficients.

What does the vertex represent on the graph?

The vertex is the lowest point of the parabola when a > 0, or the highest when a < 0. Its x-coordinate is -b/(2a), and the y-coordinate is the function evaluated there. The axis of symmetry passes vertically through it.

Can I enter fractions or decimals for the coefficients?

Both are fine. Decimals like 0.5 or -2.75 type in directly, and fractions like 3/4 or -1/2 are accepted as-is — the solver parses the slash and uses the exact ratio. Internally the math stays in full floating-point precision, so the result is accurate either way.

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