The Formula Goes Complex

When b^2 - 4ac is negative, the square root in the quadratic formula holds a negative number. In Algebra that meant no real solution; now it means two complex solutions. Take x^2 - 6x + 34 = 0: a is 1, b is -6, c is 34, and b^2 - 4ac is 36 - 136, which is -100.

So x = (6 ± √(-100))/2, and √(-100) is 10i, giving x = (6 ± 10i)/2. Divide both terms of the top by 2: x = 3 ± 5i. Split the ±: x = 3 + 5i or x = 3 - 5i.

When a, b and c are real, the two always come as a conjugate pair: the real part, b divided by 2a with its sign flipped, is shared, and the ± sits only in front of the i term.