Solve for p:
$\left(p+\frac{2}{3}\right)^2+\frac{5}{9}=0$
The given equation is \(\left(p+\frac{2}{3}\right)^2+\frac{5}{9}=0\). This is a quadratic equation where we need to isolate the variable \(p\). Notice that the term \(\left(p+\frac{2}{3}\right)^2\) represents the square of a quantity. Because the sum of this squared term and a positive constant \(\frac{5}{9}\) equals zero, we can anticipate that the solutions for \(p\) will involve complex numbers.
To solve for \(p\), we start by isolating the squared expression \(\left(p+\frac{2}{3}\right)^2\). Subtract \(\frac{5}{9}\) from both sides:
\[\left(p+\frac{2}{3}\right)^2 = -\frac{5}{9}.\]
Now, we take the square root of both sides. Since we are dealing with a negative number on the right side, we use the imaginary unit \(i\), where \(i = \sqrt{-1}\). The equation becomes:
\[p+\frac{2}{3} = \pm\sqrt{-\frac{5}{9}} = \pm i \sqrt{\frac{5}{9}} = \pm \frac{i\sqrt{5}}{3}.\]
Finally, isolate \(p\) by subtracting \(\frac{2}{3}\) from both sides of the equation:
\[p = -\frac{2}{3} \pm \frac{i\sqrt{5}}{3}.\]
Thus, the two possible values for \(p\) are:
\[p = \frac{-2 + i\sqrt{5}}{3} \quad \text{and} \quad p = \frac{-2 - i\sqrt{5}}{3}.\]
These represent the complex roots of the original quadratic equation. The final values for \(p\) are \(\boxed{-\frac{2}{3} \pm \frac{i\sqrt{5}}{3}}\).
In electrical engineering, complex numbers are fundamental to analyzing alternating current (AC) circuits. When calculating the impedance of a circuit, you often encounter quadratic equations that do not have real solutions, representing the phase shift and attenuation of signals. Solving these equations allows engineers to design robust filters and communication systems that function correctly across various frequencies.