Question

Which of the following quadratic equations has roots of 2+i and 2−i ?%0D%0A%0D%0AEquation 1: x2−4x+5=0 %0D%0A%0D%0AEquation 2: x2+4x+5=0 %0D%0A%0D%0AEquation 3: x2−4x−5=0

Answer 1

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

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

First, we calculate the sum and product of the roots \(2+i\) and \(2-i\):

1. **Sum of the roots:**
\[
r_1 + r_2 = (2+i) + (2-i) = 4
\]

2. **Product of the roots:**
\[
r_1 \cdot r_2 = (2+i)(2-i) = 2^2 - i^2 = 4 - (-1) = 4 + 1 = 5
\]

Now we can substitute these values into the quadratic equation format:

\[
x^2 - (r_1 + r_2)x + (r_1 \cdot r_2) = 0
\]
\[
x^2 - 4x + 5 = 0
\]

Now let's compare this with the given equations:

- **Equation 1:** \(x^2 - 4x + 5 = 0\) ✔️
- **Equation 2:** \(x^2 + 4x + 5 = 0\) ❌
- **Equation 3:** \(x^2 - 4x - 5 = 0\) ❌

The correct equation that has roots \(2+i\) and \(2-i\) is **Equation 1: \(x^2 - 4x + 5 = 0\)**.