The equations `x^(2) - 8x + k = 0` has real and distinct roots if :

Options

A. k = 16
B. k `gt 16`
C. `k = 8`
D. `k lt 16`

Answer

To determine the values of \( k \) for which the quadratic equation \( x^2 - 8x + k = 0 \) has real and distinct roots, we need to analyze the discriminant of the quadratic equation. The discriminant \( D \) is given by the formula:

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

For the quadratic equation \( ax^2 + bx + c = 0 \), we identify:

- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = -8 \) (coefficient of \( x \))
- \( c = k \) (constant term)

### Step 1: Calculate the Discriminant

Substituting the values of \( a \), \( b \), and \( c \) into the discriminant formula:

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

### Step 2: Simplify the Discriminant

Calculating \( (-8)^2 \):

\[ D = 64 - 4k \]

### Step 3: Set the Condition for Real and Distinct Roots

For the quadratic equation to have real and distinct roots, the discriminant must be greater than zero:

\[ 64 - 4k > 0 \]

### Step 4: Solve the Inequality

Rearranging the inequality:

\[ 64 > 4k \]

Dividing both sides by 4:

\[ 16 > k \]

or equivalently:

\[ k < 16 \]

### Conclusion

The quadratic equation \( x^2 - 8x + k = 0 \) has real and distinct roots if \( k < 16 \).