The smallest value of k, for which both the roots of the equation, `x^2-8kx + 16(k^2-k + 1)=0` are real, distinct and have values at least 4, is

Options

A. 6
B. 4
C. 2
D. 0

Explanation

To find the smallest value of \( k \) for which both roots of the equation

\[ x^2 - 8kx + 16(k^2 - k + 1) = 0 \]

are real, distinct, and have values at least 4, we can follow these steps:

### Step 1: Determine the condition for real and distinct roots
For the quadratic equation \( ax^2 + bx + c = 0 \) to have real and distinct roots, the discriminant \( D \) must be greater than 0. The discriminant is given by:

\[ D = b^2 - 4ac \]

In our case, \( a = 1 \), \( b = -8k \), and \( c = 16(k^2 - k + 1) \). Therefore, we have:

\[ D = (-8k)^2 - 4(1)(16(k^2 - k + 1)) \]

Calculating this gives:

\[ D = 64k^2 - 64(k^2 - k + 1) \]

Simplifying further:

\[ D = 64k^2 - 64k^2 + 64k - 64 = 64k - 64 \]

Setting the discriminant greater than 0 for real and distinct roots:

\[ 64k - 64 > 0 \]

Dividing by 64:

\[ k - 1 > 0 \implies k > 1 \]

### Step 2: Determine the condition for roots to be at least 4
Let the roots be \( \alpha \) and \( \beta \). For both roots to be at least 4, we can use Vieta's formulas, which state:

- The sum of the roots \( \alpha + \beta = 8k \)
- The product of the roots \( \alpha \beta = 16(k^2 - k + 1) \)

Since both roots must be at least 4, we have:

\[ \alpha + \beta \geq 8 \quad \text{(since } 4 + 4 = 8\text{)} \]

Thus, we have:

\[ 8k \geq 8 \implies k \geq 1 \]

### Step 3: Check the condition when one root is exactly 4
To ensure that at least one root is exactly 4, we can substitute \( x = 4 \) into the original equation:

\[ 4^2 - 8k(4) + 16(k^2 - k + 1) \geq 0 \]

This simplifies to:

\[ 16 - 32k + 16(k^2 - k + 1) \geq 0 \]

Expanding gives:

\[ 16 - 32k + 16k^2 - 16k + 16 \geq 0 \]

Combining like terms:

\[ 16k^2 - 48k + 32 \geq 0 \]

Dividing the entire inequality by 16:

\[ k^2 - 3k + 2 \geq 0 \]

### Step 4: Factor the quadratic inequality
Factoring gives:

\[ (k - 1)(k - 2) \geq 0 \]

### Step 5: Determine the intervals
The critical points are \( k = 1 \) and \( k = 2 \). The solution to the inequality is:

\[ k \leq 1 \quad \text{or} \quad k \geq 2 \]

### Step 6: Combine conditions
From Step 1, we found \( k > 1 \). From Step 5, we found \( k \geq 2 \). Therefore, the smallest value of \( k \) that satisfies both conditions is:

\[ \boxed{2} \]