Which of the following quadratic equations has roots of 2+i

To determine which quadratic equation has the roots \(2+i\) and \(2-i\), we can use the fact that if a quadratic has roots \(r_1\) and \(r_2\), it can be written in the form:

\[ x^2 - (r_1 + r_2)x + (r_1 \cdot r_2) = 0 \]

Where:

• \(r_1 + r_2\) is the sum of the roots
• \(r_1 \cdot r_2\) is the product of the roots

1. Calculate the sum of the roots: \[ (2+i) + (2-i) = 2 + i + 2 - i = 4 \]

2. Calculate the product of the roots: \[ (2+i)(2-i) = 2^2 - i^2 = 4 - (-1) = 4 + 1 = 5 \]

Thus, the quadratic equation can be formed as: \[ x^2 - (4)x + (5) = 0 \]

This simplifies to: \[ x^2 - 4x + 5 = 0 \]

Now we can match this with the provided options:

• Equation 1: \(x^2 - 4x + 5 = 0\) (This is the correct equation)
• Equation 2: \(x^2 + 4x + 5 = 0\) (Incorrect)
• Equation 3: \(x^2 - 4x - 5 = 0\) (Incorrect)

Therefore, the correct answer is Equation 1: \(x^2 - 4x + 5 = 0\).