Answer:
x^2 + 4x + 5 = 0
Explanation:
To determine the quadratic equation with roots x = 2 ± i, we can use the fact that complex roots always come in conjugate pairs. If one root is 2 + i, the other root will be its conjugate 2 - i.
To find the quadratic equation, we can use the fact that the sum and product of the roots of a quadratic equation are related to its coefficients.
Let's call the quadratic equation y = ax^2 + bx + c.
The sum of the roots is given by:
2 + i + 2 - i = 4
The product of the roots is given by:
(2 + i)(2 - i) = 4 - i^2 = 4 - (-1) = 5
Using these values, we can set up the equations:
Sum of roots:
b / a = 4
Product of roots:
c / a = 5
From the first equation, we can solve for b:
b = 4a
Substituting this into the second equation:
c / a = 5
We can choose any value for a, so let's set a = 1 to simplify the equation:
c = 5
Now we have the values of a, b, and c, which gives us the quadratic equation:
y = x^2 + 4x + 5 or x^2 + 4x + 5 = 0