Question

Solve the equation or write no solution. Write the solutions as integers if possible. Otherwise write them as radical expressions. $$x^{2}=-9$$

Short answer

The solutions to the equation \(x^{2}=-9\) are \(x = 3i\) and \(x = -3i\).

Step-by-step solution

Understanding the equation

The equation given is \(x^{2}=-9\). In order to solve for \(x\), one must take the square root of both sides.

Applying square root

Taking the square root of \(x^{2}\) on the left side gives us \(x\). However, taking the square root of \(-9\) on the right side is not a real number. In this case, we must use the concept of imaginary numbers. Remember that \(\sqrt{-1}\) is represented by \(i\). Hence, \(\sqrt{-9} = \sqrt{9} \cdot \sqrt{-1} = 3i\).

Writing out the solutions

The solutions to the equation are \(\pm 3i\). This means \(x = 3i\) and \(x = -3i\).