Which of the following is a solution of $x^2-2 x=-8$ ?
To find the solution to the equation \(x^2 - 2x = -8\), we first rewrite it in the standard quadratic form \(ax^2 + bx + c = 0\). Adding 8 to both sides of the equation gives:
\[x^2 - 2x + 8 = 0\]
Here, we identify the coefficients as \(a = 1\), \(b = -2\), and \(c = 8\). To solve this quadratic equation, we use the quadratic formula:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Now, we substitute the coefficients into the quadratic formula to calculate the roots:
\[x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(8)}}{2(1)}\]
Simplifying the expression:
\[x = \frac{2 \pm \sqrt{4 - 32}}{2} = \frac{2 \pm \sqrt{-28}}{2}\]
Since the value under the square root is negative, the solutions involve the imaginary unit \(i = \sqrt{-1}\). We rewrite \(\sqrt{-28}\) as \(\sqrt{28} \cdot i\). Noting that \(\sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7}\), we have:
\[x = \frac{2 \pm 2i\sqrt{7}}{2} = 1 \pm i\sqrt{7}\]
The two solutions are \(1 + i\sqrt{7}\) and \(1 - i\sqrt{7}\). Looking at the provided options, \(1 + i\sqrt{7}\) matches one of the derived roots.
Quadratic equations with complex roots appear frequently in engineering, especially in circuit analysis. When modeling an RLC circuit (Resistor-Inductor-Capacitor), the characteristic equation can lead to complex solutions, which describe the damping behavior of the circuit. For example, if the current \(I(t)\) in a circuit satisfies a differential equation resulting in \(x^2 - 2x + 8 = 0\), the complex solutions \(1 \pm i\sqrt{7}\) indicate that the system is underdamped and will oscillate as it approaches a steady state. Identifying these roots allows engineers to predict the circuit's performance accurately.