- Identify the roots \(x_1 = 2 + i\) and \(x_2 = 2 - i\).
- Calculate the sum of the roots: \(x_1 + x_2 = 4\).
- Calculate the product of the roots: \(x_1 x_2 = 5\).
- Form the quadratic equation \(x^2 - (x_1 + x_2)x + (x_1 x_2) = 0\) to get \(\boxed{x^2 - 4x + 5 = 0}\).
Explanation
- Analysis and Strategy
The objective is to find a quadratic equation that has roots \(x = 2 + i\) and \(x = 2 - i\). A quadratic equation with roots \(x_1\) and \(x_2\) can be expressed in its monic form (where the coefficient of \(x^2\) is 1) as:
\[(x - x_1)(x - x_2) = 0\]
Expanding this, we get:
\[x^2 - (x_1 + x_2)x + x_1 x_2 = 0\]
Thus, we need to calculate the sum of the roots, \(x_1 + x_2\), and the product of the roots, \(x_1 x_2\), to construct the equation.
- Calculating the Sum and Product of the Roots
We identify the roots as \(x_1 = 2 + i\) and \(x_2 = 2 - i\).
The sum of the roots is:
\[x_1 + x_2 = (2 + i) + (2 - i) = 2 + 2 + i - i = 4\]
The product of the roots is:
\[x_1 x_2 = (2 + i)(2 - i)\]
Using the difference of squares identity \((a + bi)(a - bi) = a^2 - (bi)^2 = a^2 + b^2\):
\[x_1 x_2 = 2^2 + 1^2 = 4 + 1 = 5\]
The sum is 4 and the product is 5.
- Constructing the Quadratic Equation
Now, substitute the sum \(x_1 + x_2 = 4\) and the product \(x_1 x_2 = 5\) into the monic quadratic form \(x^2 - (x_1 + x_2)x + (x_1 x_2) = 0\):
\[x^2 - (4)x + 5 = 0\]
Thus, the quadratic equation with roots \(x = 2 \pm i\) is \(x^2 - 4x + 5 = 0\). The correct equation is \(\boxed{x^2 - 4x + 5 = 0}\).
Examples
In electrical engineering, complex numbers are essential for describing alternating current (AC) circuits. When analyzing circuit resonance, the roots of the characteristic equation are often complex numbers. For example, if a circuit's impedance behavior is described by the roots \(2 \pm i\), finding the corresponding quadratic equation helps engineers determine the circuit's differential equation and predict its transient response. This demonstrates how abstract algebraic solutions underpin the design of real-world electronic systems.