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 property that for a quadratic equation in the form \( ax^2 + bx + c = 0 \), if \( r_1 \) and \( r_2 \) are the roots, then:

1. The sum of the roots \( r_1 + r_2 = -\frac{b}{a} \).
2. The product of the roots \( r_1 \cdot r_2 = \frac{c}{a} \).

Given the roots \( 2+i \) and \( 2-i \):

1. Calculate the sum of the roots:
\[
(2+i) + (2-i) = 2 + 2 = 4.
\]
Therefore, \( -\frac{b}{a} = 4 \) implies \( b = -4a \).

2. Calculate the product of the roots:
\[
(2+i)(2-i) = 2^2 - i^2 = 4 - (-1) = 5.
\]
Therefore, \( \frac{c}{a} = 5 \) implies \( c = 5a \).

Using \( a = 1 \) (standard quadratic form), we get:

- \( b = -4 \) and \( c = 5 \).

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

Now we check which of the given equations matches this form:

- **Equation 1:** \( x^2 - 4x + 5 = 0 \) (this is the one we found).
- **Equation 2:** \( x^2 + 4x + 5 = 0 \) (does not match).
- **Equation 3:** \( x^2 - 4x - 5 = 0 \) (does not match).

Thus, the equation with roots \( 2+i \) and \( 2-i \) is **Equation 1: \( x^2 - 4x + 5 = 0 \)**.