The letter i
The symbol i is the imaginary unit. By definition, i² = −1. That single rule is enough to finish the quadratic formula when b² − 4ac is negative. The square root of a negative number is written as i times the square root of the matching positive number: √(−16) = 4i, because (4i)² = 16 · i² = −16.
A complex number in rectangular form is a real part plus an imaginary part. The number −1 + 2i has real part −1 and imaginary part 2. The number 3 is already complex: its imaginary part is 0. The calculator prints a + bi only when the imaginary part is not zero, and it prints an ordinary real number when Δ is not negative.
Why the roots come in a pair
If the coefficients a, b, and c are real, a non-real root never appears alone. Its conjugate — the same real part and the opposite imaginary part — is also a root. That is why x² + 2x + 5 = 0 produces both −1 + 2i and −1 − 2i. You can check the pair by expanding (x − (−1 + 2i))(x − (−1 − 2i)). The imaginary pieces cancel, and you recover a quadratic with real coefficients.
The quadratic formula calculator stops at a negative discriminant because that page stays on the real line. The complex page continues the same formula. The only extra step is writing √Δ as i√|Δ|.
A short substitution
For x² + 2x + 5 = 0 the coefficients are a = 1, b = 2, and c = 5. Then Δ = 4 − 20 = −16. The quadratic formula gives x = (−2 ± √(−16)) / 2 = (−2 ± 4i) / 2 = −1 ± 2i. Those are the same numbers the calculator prints. If you change c to 1, Δ becomes 0 and the root is the real number −1. If you change c to −3, Δ is 16 and the roots are the real numbers 1 and −3.
What the page does not claim
The calculator does not plot a complex plane, does not convert a + bi into polar form, and does not solve cubics. It also does not treat complex coefficients. The word “complex” here means “a real part and an imaginary part,” not “complicated.” A complex root is a complete answer to the quadratic; it is not a leftover error term.
If you only need to know whether the parabola crosses the x-axis, the sign of Δ is enough. Crossing requires a non-negative discriminant. A negative discriminant means the graph stays above or below the axis, while the algebra still has two roots off the real line.