The equation `x^(2) - 4x + 29 =0` has one root 2+ 5i. What is the other root?

Options

A. 2
B. 5
C. ` 2 + 5 i `
D. ` 2 - 5 i `

Explanation

To find the other root of the quadratic equation \(x^2 - 4x + 29 = 0\) given that one root is \(2 + 5i\), follow these steps:

### Step 1: Understand the nature of roots
In a quadratic equation with real coefficients, complex roots always occur in conjugate pairs. This means if one root is \(a + bi\), the other root must be \(a - bi\).

### Step 2: Identify the given root
The given root is \(2 + 5i\). Here, \(a = 2\) and \(b = 5\).

### Step 3: Find the conjugate of the given root
To find the other root, we take the conjugate of the given root:
- The conjugate of \(2 + 5i\) is \(2 - 5i\).

### Step 4: State the other root
Thus, the other root of the equation \(x^2 - 4x + 29 = 0\) is \(2 - 5i\).

### Final Answer
The other root is \(2 - 5i\).

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