Question

The quadratic equation $ x^2 - 5x + k = 0 $ has equal roots. Find the value of $ k $.

• A. \( 6 \)
• B. \( \frac{25}{4} \)
• C. \( \frac{9}{4} \)
• D. \( 0 \)

Hint:

Tip: The discriminant condition is crucial for determining the nature of roots in quadratic equations.

Answer 1

The Correct Option is B

The problem involves finding the value of \( k \) for a quadratic equation with equal roots. A quadratic equation \( ax^2 + bx + c = 0 \) has equal roots when the discriminant \( \Delta = b^2 - 4ac \) is zero. Given the equation \( x^2 - 5x + k = 0 \), here \( a = 1 \), \( b = -5 \), and \( c = k \). Therefore, the discriminant is:

\(\Delta = (-5)^2 - 4 \cdot 1 \cdot k = 0\)

This simplifies to:

\(25 - 4k = 0\)

Solving for \( k \), we get:

\(4k = 25\)

\(k = \frac{25}{4}\)

Thus, the value of \( k \) that ensures the quadratic equation has equal roots is \(\frac{25}{4}\).

Answer 2

Step 1: Condition for equal roots 
For equal roots, the discriminant \( \Delta = b^2 - 4ac = 0 \).

Step 2: Apply values 
Here, \( a = 1 \), \( b = -5 \), \( c = k \). \[ (-5)^2 - 4 \times 1 \times k = 0 \Rightarrow 25 - 4k = 0 \]

Step 3: Solve for \( k \) 
\[ 4k = 25 \Rightarrow k = \frac{25}{4} \]