Which of the following quadratic functions has complex roots $x=-2+3 i$ and $x=-2-3 i$?
To determine which quadratic function has roots \(x = -2 + 3i\) and \(x = -2 - 3i\), we can use the relationship between the roots of a quadratic equation and its coefficients. A quadratic equation with roots \(r_1\) and \(r_2\) can be written in the form \(a(x - r_1)(x - r_2) = 0\). For a monic quadratic (where the leading coefficient is 1), this simplifies to \(x^2 - (r_1 + r_2)x + r_1r_2 = 0\). Here, the given roots are \(r_1 = -2 + 3i\) and \(r_2 = -2 - 3i\).
First, we calculate the sum of the roots:
\[S = r_1 + r_2 = (-2 + 3i) + (-2 - 3i) = -2 - 2 + 3i - 3i = -4.\]
Next, we calculate the product of the roots:
\[P = r_1 \times r_2 = (-2 + 3i)(-2 - 3i).\]
Using the difference of squares identity, \((a+b)(a-b) = a^2 - b^2\), we have:
\[P = (-2)^2 - (3i)^2 = 4 - 9i^2.\]
Since \(i^2 = -1\), the product is:
\[P = 4 - 9(-1) = 4 + 9 = 13.\]
With the sum \(S = -4\) and the product \(P = 13\), the quadratic function is given by:
\[f(x) = x^2 - Sx + P = x^2 - (-4)x + 13 = x^2 + 4x + 13.\]
Comparing this to the given options:
We can see that the correct function is option A.
In engineering and physics, complex numbers are often used to describe oscillating systems, such as electrical circuits or mechanical vibrations. For example, if you are modeling a damped harmonic oscillator, the roots of the characteristic equation provide information about the system's frequency and decay rate. Finding the quadratic function from these roots is a fundamental skill in analyzing these systems, ensuring that you can construct the correct governing equations for the dynamics of your model.