Verified Answer :

  • Identify the sum of the roots: \(x_1 + x_2 = -3\).
  • Calculate the product of the roots: \(x_1 \cdot x_2 = 3\).
  • Substitute these into the standard quadratic form \(x^2 - (\text{sum})x + (\text{product}) = 0\), which results in \(x^2 + 3x + 3 = 0\).
  • The final equation is \(\boxed{d}\).

Explanation

  1. Problem Analysis and Setup

To determine the quadratic equation with the roots \(x = \frac{-3 \pm \sqrt{3} i}{2}\), we recall that any quadratic equation with roots \(x_1\) and \(x_2\) can be expressed as \(a(x - x_1)(x - x_2) = 0\). For a monic quadratic equation (where the coefficient of \(x^2\) is \(1\)), this simplifies to \(x^2 - (x_1 + x_2)x + (x_1 \cdot x_2) = 0\), where \(x_1 + x_2\) is the sum of the roots and \(x_1 \cdot x_2\) is their product. Given our specific roots \(x_1 = \frac{-3 + \sqrt{3}i}{2}\) and \(x_2 = \frac{-3 - \sqrt{3}i}{2}\), we can calculate these values.

  1. Calculating the Sum and Product of Roots

First, we calculate the sum of the roots:
\[x_1 + x_2 = \left(\frac{-3 + \sqrt{3}i}{2}\right) + \left(\frac{-3 - \sqrt{3}i}{2}\right) = \frac{-3 + \sqrt{3}i - 3 - \sqrt{3}i}{2} = \frac{-6}{2} = -3.\]
Next, we calculate the product of the roots using the difference of squares identity \((A-B)(A+B) = A^2 - B^2\):
\[x_1 \cdot x_2 = \left(\frac{-3 + \sqrt{3}i}{2}\right) \cdot \left(\frac{-3 - \sqrt{3}i}{2}\right) = \frac{(-3)^2 - (\sqrt{3}i)^2}{4}.\]
Since \((\sqrt{3}i)^2 = 3 \cdot i^2 = 3 \cdot (-1) = -3\), the product becomes:
\[x_1 \cdot x_2 = \frac{9 - (-3)}{4} = \frac{12}{4} = 3.\]

  1. Conclusion and Identifying the Equation

Now, we substitute the sum and product into the monic quadratic equation form \(x^2 - (x_1 + x_2)x + (x_1 \cdot x_2) = 0\):
\[x^2 - (-3)x + 3 = 0 \implies x^2 + 3x + 3 = 0.\]
Comparing this result with the given choices:

A\(x^2 + 3x + 12 = 0\)
B\(2x^2 + 6x + 9 = 0\)
C\(2x^2 + 6x + 3 = 0\)
D\(x^2 + 3x + 3 = 0\)

We can see that the resulting equation exactly matches option (d).

Examples

Quadratic equations with complex roots are frequently encountered in electrical engineering when analyzing RLC circuits (Resistor-Inductor-Capacitor circuits). The characteristic equation of such a circuit often results in complex roots, where the real part indicates the damping factor (the rate at which oscillations die out) and the imaginary part relates to the frequency of oscillation. By identifying the correct quadratic equation from its roots, engineers can design circuits with the desired stability and performance characteristics.

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