L^2=-9
You asked:
Solve the equation \({L}^{2} = -9\) for the variable \(L\).
MathBot Answer:
\[\begin{aligned}L &= 3 i \approx 3.0 i\\L &= - 3 i \approx - 3.0 i\end{aligned}\]
Take the square root of both sides of the equation $$\begin{aligned}L^{2} &= -9 \\ \sqrt{L^{2}} &= \sqrt{-9} \end{aligned}$$
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 \(i\). $$\begin{aligned}L &= \sqrt{9} \cdot \sqrt{-1}\\L &= \pm 3 i \\ L = 3 i &,\, L=-3 i\end{aligned}$$
\(i\) is the imaginary unit, defined as \(i^2 = -1\).