If equation `x^(2) + kx + 64 = 0 and x^(2) - 8x + k = 0` have real roots. Then what is the value of k ?

Options

A. 4
B. 8
C. 12
D. 16

Answer

To find the value of \( k \) such that the equations \( x^2 + kx + 64 = 0 \) and \( x^2 - 8x + k = 0 \) have real roots, we will use the discriminant condition for real roots. The discriminant \( D \) must be greater than or equal to zero.

### Step 1: Analyze the first equation
The first equation is:
\[ x^2 + kx + 64 = 0 \]
Here, \( a = 1 \), \( b = k \), and \( c = 64 \). The discriminant \( D_1 \) for this equation is given by:
\[ D_1 = b^2 - 4ac = k^2 - 4(1)(64) = k^2 - 256 \]
For the roots to be real, we need:
\[ D_1 \geq 0 \implies k^2 - 256 \geq 0 \]
This simplifies to:
\[ k^2 \geq 256 \]
Taking the square root of both sides, we get:
\[ k \geq 16 \quad \text{or} \quad k \leq -16 \]

### Step 2: Analyze the second equation
The second equation is:
\[ x^2 - 8x + k = 0 \]
Here, \( a = 1 \), \( b = -8 \), and \( c = k \). The discriminant \( D_2 \) for this equation is:
\[ D_2 = b^2 - 4ac = (-8)^2 - 4(1)(k) = 64 - 4k \]
For the roots to be real, we need:
\[ D_2 \geq 0 \implies 64 - 4k \geq 0 \]
This simplifies to:
\[ 64 \geq 4k \implies 16 \geq k \]

### Step 3: Combine the inequalities
From the first equation, we have:
\[ k \geq 16 \quad \text{or} \quad k \leq -16 \]
From the second equation, we have:
\[ k \leq 16 \]
Combining these inequalities, we find:
\[ k = 16 \]

### Conclusion
Thus, the value of \( k \) such that both equations have real roots is:
\[ \boxed{16} \]