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:
- The sum of the roots \( r_1 + r_2 = -\frac{b}{a} \).
- The product of the roots \( r_1 \cdot r_2 = \frac{c}{a} \).
Given the roots \( 2+i \) and \( 2-i \):
-
Calculate the sum of the roots: \[ (2+i) + (2-i) = 2 + 2 = 4. \] Therefore, \( -\frac{b}{a} = 4 \) implies \( b = -4a \).
-
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: \( x^2 - 4x + 5 = 0 \).