Chapter 4.4, Problem 3AGP

To determine

Find the solution of the given equation.

Answer to Problem 3AGP

The solution of the equation is $x=\pm 5i$ .

Explanation of Solution

Given:

The given equation is $4x^2+100=0$ .

Calculation:

  $4x^2+100=0$

Subtract $100$ from both sides.

  $\begin{array}{l}4x^2+100-100=0-100\\ \Rightarrow 4x^2=-100\end{array}$

Divide both sides by $4$ .

  $\begin{array}{l}\Rightarrow \frac{4}{4}{x}^2=-\frac{100}{4}\\ \Rightarrow x^2=-25\end{array}$

Apply square root property.

  $\Rightarrow x=\pm \sqrt{-25}$

Pure imaginary number $\to$ it is a square root of negative real number. For any positive real number $b$ , $\sqrt{-b^2}=\sqrt{b^2}\cdot \sqrt{-1}\text{or}bi$ . The number $i$ is the principal square root of $-1$ ; that is, $i=\sqrt{-1}$ .

  $\begin{array}{l}\Rightarrow x=\pm \sqrt{-5^2}\\ \Rightarrow x=\pm 5i\end{array}$

Hence the solution of the equation is $x=\pm 5i$ .