Free Online Quadratic Equation Solver
Solve quadratic equations with steps and imaginary roots.
The equation is ax² + bx + c = 0. Decimals and simple fractions such as
1/2 are accepted. Leave a field empty to treat it as zero.
Step-by-step working
- Enter a, b, and c to see the working.
About the Quadratic Equation Solver
A quadratic equation has the form ax² + bx + c = 0, where a is not zero. Its graph is a parabola, and solving the equation means finding the x-values where that parabola crosses the x-axis. This solver reports the discriminant, the real or complex roots, the vertex, the axis of symmetry, the factored form, and the y-intercept, with the arithmetic shown step by step.
The discriminant decides the type of roots
The discriminant is Δ = b² − 4ac. Its sign tells you what to expect before you finish the quadratic formula. When Δ is positive, the parabola crosses the x-axis at two distinct real points. When it is zero, the parabola touches the axis at one point, so the equation has a repeated real root. When it is negative, no real number squares to a negative value, so the two roots are complex conjugates that involve i. The discriminant is also the expression under the square root, which is why its sign controls the whole calculation.
How the quadratic formula is applied
The quadratic formula is x = (−b ± √Δ) / (2a). The calculator substitutes the coefficients, evaluates the discriminant, simplifies the square root when possible, and then separates the plus and minus cases. When the discriminant is not a perfect square, the exact answer contains a surd, such as (3 ± √17) / 2, while the decimal result is rounded for readability. When the discriminant is negative, the square root becomes an imaginary number and the result is written as a real part plus or minus an imaginary part.
Vertex, axis, and factored form
The vertex of the parabola sits at x = −b / (2a). Substituting that x back into the equation gives the y-coordinate. The vertical line through the vertex is the axis of symmetry, and it always passes halfway between the two real roots. If real roots exist, the expression can be rewritten as a(x − r₁)(x − r₂). A negative discriminant means the quadratic cannot be factored into real linear factors, although it can still be factored over the complex numbers.
When a is zero
If a is zero the equation is no longer quadratic. The solver detects this case and solves the linear equation bx + c = 0 instead. If b is also zero, the result is either no solution or infinitely many solutions, depending on c. Quadratic-specific values such as the discriminant and vertex are left blank because they do not apply to a straight line.
Frequently asked questions
Why do I get two answers?
Every quadratic can have up to two roots because the parabola can cross the horizontal axis twice. The plus and minus signs in the quadratic formula produce the two values. A repeated root counts as one value with multiplicity two.
What does a negative discriminant mean in practice?
The graph never crosses the x-axis, so there is no real solution. The roots are complex numbers and are useful in engineering, signal processing, and advanced algebra, even though they cannot be plotted as ordinary x-intercepts.
Can I enter fractions instead of decimals?
Yes. Inputs such as 1/2 or -3/4 are evaluated before solving. This is handy when the coefficients come from a textbook problem that uses exact fractions rather than decimals.