Quadratic Formula Calculator
Solve quadratic equations with discriminant and roots
About the Quadratic Formula Calculator
The quadratic formula solves any equation of the form ax² + bx + c = 0, and this calculator applies it in full — including the cases where the answers are repeated or complex. Enter the three coefficients and it returns the roots, the discriminant, the vertex, and the axis of symmetry.
It is a constant companion in algebra and precalculus, where solving quadratics and analyzing parabolas is bread-and-butter work. By computing the discriminant first, it correctly identifies whether you get two real roots, one repeated root, or a complex conjugate pair — and it shows the complex answers properly rather than just failing when the discriminant is negative.
How to Use the Quadratic Formula Calculator
- Enter coefficient a (it cannot be zero, or the equation is not quadratic).
- Enter coefficients b and c.
- Read the roots, with complex roots shown in proper a + bi form when the discriminant is negative.
- Check the discriminant, vertex, and axis of symmetry, along with the worked formula.
Why Use ToolForge’s Quadratic Formula Calculator
- It handles all three discriminant cases — two real roots, one repeated root, and complex conjugate roots — instead of breaking on a negative discriminant.
- It computes the vertex and axis of symmetry too, so you can analyze the parabola, not just solve for x.
- It shows the discriminant and the full formula substitution, making the method clear.
- It runs locally with no account.
Frequently Asked Questions
What is the discriminant and why does it matter?
The discriminant is b² − 4ac, the part under the square root in the quadratic formula. Its sign tells you the nature of the roots: positive gives two real roots, zero gives one repeated root, and negative gives a pair of complex roots.
What happens when the discriminant is negative?
The equation has no real solutions, but it has two complex conjugate roots. This calculator computes them and displays them in the form a + bi rather than simply reporting "no solution".
What are the vertex and axis of symmetry?
The vertex is the turning point of the parabola the equation describes, and the axis of symmetry is the vertical line through it, at x = −b/(2a). The calculator reports both so you can sketch or analyze the curve.
Why must coefficient a be non-zero?
If a is zero, the x² term disappears and the equation is linear, not quadratic — and the formula would divide by zero. The calculator requires a non-zero a; for a linear equation, solve bx + c = 0 directly instead.
