- Rewrite the quadratic equation in standard form: \(x^2 - 16x + 65 = 0\).
- Apply the quadratic formula \(x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}\).
- Simplify the discriminant \(-4\) to obtain the complex solutions \(x = 8 \pm i\).
- The final solutions are \(\boxed{x = 8 + i, x = 8 - i}\).
Explanation
- Standardizing the Quadratic Equation
We are given the quadratic equation \(x^2 = 16x - 65\). To find the solutions, we first rewrite the equation in the standard form \(Ax^2 + Bx + C = 0\). Subtracting \(16x\) and adding \(65\) to both sides, we obtain:
\[x^2 - 16x + 65 = 0\]
Here, the coefficients are \(A = 1\), \(B = -16\), and \(C = 65\).
- Applying the Quadratic Formula
To solve for \(x\), we apply the quadratic formula:
\[x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}\]
Substituting the values \(A = 1\), \(B = -16\), and \(C = 65\) into the formula:
\[x = \frac{-(-16) \pm \sqrt{(-16)^2 - 4(1)(65)}}{2(1)}\]
\[x = \frac{16 \pm \sqrt{256 - 260}}{2}\]
\[x = \frac{16 \pm \sqrt{-4}}{2}\]
- Simplifying the Solutions
The expression under the square root is \(-4\), which is negative. Since \(\sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i\), where \(i\) is the imaginary unit, we can simplify the expression:
\[x = \frac{16 \pm 2i}{2}\]
Dividing both terms by \(2\), we get:
\[x = 8 \pm i\]
Thus, the solutions are \(x = 8 + i\) and \(x = 8 - i\). Comparing these to the required form \(a \pm bi\), we identify \(a = 8\) and \(b = 1\).
Examples
Quadratic equations with complex roots appear frequently in electrical engineering, specifically when analyzing RLC circuits (Resistor-Inductor-Capacitor). When the circuit is underdamped, the characteristic equation often leads to complex roots, which represent oscillations that decay over time. Understanding how to find these roots allows engineers to predict the behavior of signals in communication systems and power grids.