How this instrument works
A quadratic ax² + bx + c = 0 has two solutions given by x = (−b ± √(b² − 4ac)) ⁄ 2a. Most of the time the term under the root is positive and the two roots sit on the number line. But when b² − 4ac is negative, that square root has no real value — and rather than stopping there, algebra pushes through by writing it as i times the square root of the positive quantity 4ac − b². Split apart, that gives exactly the two fields this instrument reports: Re = −b ⁄ 2a stays the same for both roots, while Im = √(4ac − b²) ⁄ 2a attaches once with a plus sign and once with a minus.
The reason one real part serves both roots is not a coincidence of this particular equation — it is forced by a, b, and c all being real numbers. Take the equation az² + bz + c = 0 and apply complex conjugation to every term: since conjugating a real number leaves it unchanged, the conjugated equation reads a z̄² + b z̄ + c = 0, which says z̄ solves the same quadratic whenever z does. Two roots, one conjugate relationship, no exceptions — this is why a complex root's partner is never a free choice the way it is when you conjugate an arbitrary complex number by hand.
Geometrically, a negative discriminant means the parabola y = ax² + bx + c never touches the x-axis at all — it stays entirely above it or entirely below it, and the two complex roots are the mirror-image points where it 'would have' crossed, reflected across the real axis at height Im and depth −Im. The boundary case is worth knowing: as b² − 4ac rises to exactly zero, Im shrinks to zero and the two complex roots merge into one repeated real root sitting exactly on the axis — the tangent case, not a complex one at all.
- Enter the quadratic's coefficients into a (coefficient of x²), b (coefficient of x), and c (constant), matching ax² + bx + c = 0 term for term.
- Confirm b² − 4ac lands negative — if it does not, the equation has real roots instead and this sheet will say so rather than guess.
- Read Real part of root: one figure, shared by both roots of the equation.
- Read Imaginary part of root (± ): pair it once with a plus and once with a minus to write out both roots in full.
Worked example — x² + 2x + 5 = 0
Set a = 1, b = 2, c = 5. First check the discriminant: b² − 4ac = 4 − 20 = −16, which is negative, so the two roots are genuinely complex rather than real. Re = −b ⁄ 2a = −2 ⁄ 2 = −1. Im = √(4ac − b²) ⁄ 2a = √(20 − 4) ⁄ 2 = √16 ⁄ 2 = 4 ⁄ 2 = 2. The instrument reports re = −1 and im = 2, meaning the two roots are −1 + 2i and −1 − 2i.
A quick check confirms it: multiply the two roots back out as (x − (−1 + 2i))(x − (−1 − 2i)) = (x + 1 − 2i)(x + 1 + 2i) = (x + 1)² − (2i)² = x² + 2x + 1 + 4 = x² + 2x + 5, the exact equation entered. That the imaginary terms cancel completely is not luck — it is the conjugate pair doing what conjugate pairs always do, closing the product back into real numbers.
Questions
Why do complex roots of a quadratic always come in conjugate pairs?
Because a, b, and c are real numbers. Conjugating the whole equation az² + bz + c = 0 leaves every coefficient unchanged, so if z is a root, its conjugate z̄ automatically satisfies the same equation. There is no scenario with real coefficients where a lone, unpaired complex root exists.
How is this different from just conjugating a complex number by hand?
Conjugating an arbitrary complex number is a choice — you flip the sign of whatever imaginary part it happens to have. Here the flip is not a choice at all: it is forced on the second root purely because the quadratic's coefficients are real. This calculator hands you the pair as a consequence of that constraint, not as an operation you apply.
What does a negative discriminant mean on the graph of the parabola?
It means y = ax² + bx + c never crosses the x-axis — the whole curve sits above it or below it. The complex roots mark where the curve would have crossed if the axis dipped down (or rose up) to meet it; their imaginary part measures how far short the parabola falls.
Why does the formula use √(4ac − b²) instead of the usual √(b² − 4ac)?
Because b² − 4ac is negative in this situation, its square root has no real value. Flipping the sign to 4ac − b² makes the quantity under the root positive, so an ordinary real square root works, and the factor of i is written outside it separately. Both expressions describe the same discriminant, just rearranged so the arithmetic stays real until the last step.
What happens if the discriminant is exactly zero or positive instead?
A discriminant of exactly zero gives Im = 0 and a single repeated real root sitting on the x-axis — a tangent parabola, not a complex pair. A positive discriminant gives two distinct real roots. This instrument is built for the negative case; feeding it a non-negative discriminant is a sign the equation does not need complex numbers at all.