Solve the equation using the quadratic formula.
$x^2-12 x+45=0$
To solve the quadratic equation \(x^2 - 12x + 45 = 0\), we identify the coefficients by comparing it to the standard form \(ax^2 + bx + c = 0\). Here, we have \(a = 1\), \(b = -12\), and \(c = 45\). The quadratic formula to find the roots is given by \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.\]
Next, we calculate the discriminant \(D = b^2 - 4ac\), which determines the nature of the roots. Substituting our values: \[D = (-12)^2 - 4(1)(45) = 144 - 180 = -36.\] Since the discriminant is negative, the equation has two complex conjugate roots. We will use the imaginary unit \(i = \sqrt{-1}\) to express these roots, noting that \(\sqrt{-36} = \sqrt{36} \cdot \sqrt{-1} = 6i\).
Now we substitute the values back into the quadratic formula: \[x = \frac{-(-12) \pm 6i}{2(1)} = \frac{12 \pm 6i}{2}.\] Simplifying this expression by dividing both parts of the numerator by 2, we obtain: \[x = 6 \pm 3i.\] Therefore, the roots are \(x = 6 + 3i\) and \(x = 6 - 3i\). The solution set for the given quadratic equation is \(\{6 + 3i, 6 - 3i\}\).
Quadratic equations with complex roots appear frequently in electrical engineering when analyzing AC circuits. For instance, in an RLC (resistor-inductor-capacitor) circuit, the behavior of the current is described by a differential equation that leads to a characteristic quadratic equation. If the discriminant is negative, it indicates an underdamped system where the current oscillates with a decaying envelope. Solving such equations, like \(x^2 - 12x + 45 = 0\), allows engineers to predict the frequency and damping factor of these oscillations, which is essential for designing stable electronics.