What does the discriminant tell me?

The discriminant (b² − 4ac) determines the type of roots: positive means two distinct real roots, zero means one repeated real root (a double root), and negative means two complex conjugate roots.

What are complex roots, and why do they come in pairs?

When the discriminant is negative, its square root isn't a real number — it's expressed using i (the imaginary unit, where i² = −1). For a quadratic with real coefficients, complex roots always appear as a conjugate pair: real ± imaginary·i.

What happens if I enter a = 0?

The equation is no longer quadratic — with a = 0 it becomes linear (bx + c = 0), which the calculator solves separately as x = −c/b, or flags as having no solution / infinite solutions if b is also 0.

Can a quadratic equation have no real solutions?

Yes — whenever the discriminant is negative, there are no real solutions, only a complex conjugate pair. Graphically, this means the parabola never crosses the x-axis.

How is the vertex of the parabola related to this?

The vertex's x-coordinate is always −b/2a — notice this is the same term the ± in the quadratic formula is added to and subtracted from, which is why two real roots are always symmetric around the vertex.

Does the order of a, b, c matter?

Yes — they must correspond to the standard form ax² + bx + c = 0. Entering coefficients from a differently-arranged equation (e.g. one not yet set to equal 0) will give an incorrect answer unless you rearrange it first.

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Math & Everyday

Quadratic Equation Solver

Real or complex roots of ax² + bx + c = 0, with the discriminant shown.

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About the Quadratic Equation Solver

Solves any quadratic equation ax² + bx + c = 0 for x, using the quadratic formula, and reports whether the roots are two distinct real numbers, one repeated real number, or a complex conjugate pair, depending on the discriminant.

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How to use it

Enter the coefficients a, b, and c from your equation in the form ax² + bx + c = 0.
Read the roots in the result panel, along with the discriminant.

Formula

x = (−b ± √(b² − 4ac)) ÷ 2a

The discriminant, b² − 4ac, determines the nature of the roots: positive gives two distinct real roots, zero gives one repeated real root, and negative gives two complex conjugate roots (expressed as real ± imaginary·i, since the square root of a negative number isn't a real number).

Worked example

For x² − 5x + 6 = 0 (a=1, b=−5, c=6): discriminant = (−5)² − 4(1)(6) = 25 − 24 = 1, which is positive, giving two real roots: x = 3 and x = 2.

Discriminant → Root Type

Discriminant → Root Type
Discriminant (b²−4ac)	Root Type	Example
> 0	Two distinct real roots	x²−5x+6=0 → x=2, 3
= 0	One repeated real root	x²−4x+4=0 → x=2
< 0	Two complex conjugate roots	x²+x+1=0 → x=−0.5±0.866i

Interpreting your result

If a = 0, the equation isn't actually quadratic — it reduces to a linear equation (bx + c = 0), or, if b is also 0, either an identity (always true, if c = 0 too) or a contradiction (never true). The calculator detects these edge cases and explains them rather than silently dividing by zero.

Recommendations

• A positive discriminant always means two distinct real roots; check its sign first to know what kind of answer to expect before looking at the roots themselves.
• A discriminant of exactly 0 means the parabola touches the x-axis at exactly one point (a repeated root) — the vertex sits exactly on the axis.
• Complex roots always come in conjugate pairs (real ± imaginary·i) for a quadratic with real coefficients — you'll never get just one complex root alone.

Frequently asked questions

x = (−b ± √(b² − 4ac)) ÷ 2a — it gives the value(s) of x that satisfy any equation of the form ax² + bx + c = 0, provided a ≠ 0.