Question

Which of the following quadratic equations has roots of 2+i
and 2−i
?

Equation 1: x2−4x+5=0

Equation 2: x2+4x+5=0

Equation 3: x2−4x−5=0

(1 point)
Equation

Answer 1

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: \(x^2 - 4x + 5 = 0\)**.