Squares Equal to Negatives

In Algebra, ${x}^{2} = -9$ had no solution, because no real number squares to -9. With i it has two. Start from ${x}^{2}+9 = 0$: move the 9 across to get ${x}^{2} = -9$, then take the square root of both sides: $x = 3i$ or $x = -3i$.

Why two? Both ${\left(3i\right)}^{2}$ and ${\left(-3i\right)}^{2}$ equal $9{i}^{2}$, which is -9, just as both 3 and -3 square to 9. The square root of both sides always brings the ± with it.

The two solutions are conjugates of each other, and that is no accident: when a quadratic with real a, b and c has a solution with i, the conjugate is the other solution.