Quadratic Formula Calculator · Practical guide
Quadratic Equation with No Real Solutions
When b² − 4ac is negative, the parabola never crosses the x-axis and the roots come out as complex conjugates.
At a glance
Quick answer
A quadratic equation has no real solutions when its discriminant is negative: D = b² − 4ac < 0. Instead, it has two complex conjugate roots in the form p ± qi. For example, x² + 2x + 5 = 0 has D = 4 − 20 = −16, so x = (−2 ± 4i) / 2 = −1 ± 2i. Enter the coefficients above and the calculator shows the real and imaginary parts directly.
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Solution
Step-by-Step:
Worked Examples with Complex Roots
| Equation | Discriminant | Complex roots |
|---|---|---|
| x² + 2x + 5 = 0 | −16 | −1 ± 2i |
| x² + x + 1 = 0 | −3 | −0.5 ± 0.8660i |
| x² + 9 = 0 | −36 | ±3i |
| 3x² + 2x + 1 = 0 | −8 | −0.3333 ± 0.4714i |
How the Roots Come Out
Take x² + x + 1 = 0, with a = 1, b = 1, c = 1. The discriminant is D = 1 − 4 = −3, so √D = √3 · i ≈ 1.7321i. The formula gives x = (−1 ± 1.7321i) / 2 = −0.5 ± 0.8660i. The real part is always −b / 2a — the same x-coordinate as the vertex — and the imaginary part is √|D| / 2a.
What It Means Graphically
Real roots are x-intercepts, so no real roots means the parabola never touches the x-axis. For x² + 2x + 5, the vertex sits at (−1, 4) and the parabola opens upward, hovering entirely above the axis with a minimum value of 4. Flip the sign of a and the whole curve hangs below the axis instead.
Checking a Complex Answer
Substitution still works: plug x = −1 + 2i into x² + 2x + 5. Since (−1 + 2i)² = 1 − 4i + 4i² = −3 − 4i (because i² = −1), the full expression becomes (−3 − 4i) + (−2 + 4i) + 5 = 0. The imaginary parts cancel — they always do for conjugate roots, which is why the pair solves the equation together.
Frequently Asked Questions
What does the i in the answer mean?
The letter i is the imaginary unit, defined by i² = −1. It lets you take the square root of a negative discriminant: √−16 = 4i.
Why do complex roots always come in pairs?
Because the ± in the quadratic formula produces two roots that differ only in the sign of the imaginary part. If p + qi is a root of a real-coefficient quadratic, p − qi must be the other.
Can a quadratic have exactly one real root and one complex root?
No. With real coefficients, the discriminant is either zero (one repeated real root), positive (two real roots), or negative (two complex roots) — mixing is impossible.
Where are complex roots used in real life?
They appear in electrical engineering (AC circuit impedance), signal processing, and physics whenever oscillations or waves are modeled — anywhere a system cycles rather than crossing zero.