What are the solutions of $2 x^2+8 x+26=0$ ?
To solve the quadratic equation \(2x^2 + 8x + 26 = 0\), we first look for ways to simplify the expression. All terms in the equation are divisible by 2, so dividing the entire equation by 2 yields:
\[x^2 + 4x + 13 = 0\]
This simplifies our calculations significantly. We now identify the coefficients as \(a = 1\), \(b = 4\), and \(c = 13\).
To find the roots of the quadratic equation \(x^2 + 4x + 13 = 0\), we use the quadratic formula:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Substituting the identified coefficients \(a = 1\), \(b = 4\), and \(c = 13\) into the formula:
\[x = \frac{-4 \pm \sqrt{4^2 - 4(1)(13)}}{2(1)}\]
\[x = \frac{-4 \pm \sqrt{16 - 52}}{2}\]
\[x = \frac{-4 \pm \sqrt{-36}}{2}\]
Since we have a negative value under the square root, the roots will involve the imaginary unit \(i\), where \(i = \sqrt{-1}\). Thus, \(\sqrt{-36} = \sqrt{36} \cdot i = 6i\).
\[x = \frac{-4 \pm 6i}{2}\]
\[x = -2 \pm 3i\]
Thus, the solutions are \(x = -2 + 3i\) and \(x = -2 - 3i\).
In electrical engineering, quadratic equations with negative discriminants frequently appear when analyzing RLC circuits (Resistor, Inductor, Capacitor). The roots determine the oscillation frequency and the damping factor of the circuit. Solving \(x^2 + 4x + 13 = 0\) might represent a circuit's natural response where the imaginary part \(\pm 3i\) determines the frequency of oscillation, and the real part \(-2\) indicates the rate at which the signal decays over time.