x^2=-1
x=?
You asked:
Solve the equation \({x}^{2} = -1\) for the variable \(x\).
MathBot Answer:
\[\begin{aligned}x &= i \approx 1.0 i\\x &= - i \approx - 1.0 i\end{aligned}\]
Take the square root of both sides of the equation $$\begin{aligned}x^{2} &= -1 \\ \sqrt{x^{2}} &= \sqrt{-1} \end{aligned}$$
Remember that both positive and negative numbers will result in a positive number when squared. This means that the square root of a negative number is undefined over the real numbers. To resolve this, the imaginary unit, \(i\) is defined as \(\sqrt{-1}\). A number containing the imaginary unit is called an imaginary number. Square roots of negative numbers can thus be expressed in terms of \(i\). $$\begin{aligned}x &= \sqrt{1} \cdot \sqrt{-1}\\x &= \pm i \\ x = i &,\, x=-i\end{aligned}$$
\(i\) is the imaginary unit, defined as \(i^2 = -1\).