Question

If the roots of quadratic equation \(4x^2 - 5x + k = 0\) are real and equal, then value of \(k\) is:

• A. \(\frac{5}{4}\)
• B. \(\frac{25}{16}\)
• C. \(-\frac{5}{4}\)
• D. \(-\frac{25}{16}\)

Answer 1

The Correct Option is B

Step 1: Understanding the condition for real and equal roots:
For a quadratic equation \( ax^2 + bx + c = 0 \), the discriminant \( \Delta \) is given by:
\[ \Delta = b^2 - 4ac \] The roots of the quadratic equation are real and equal if and only if the discriminant is zero, i.e., \( \Delta = 0 \).

Step 2: Apply the condition to the given quadratic equation:
The given quadratic equation is \( 4x^2 - 5x + k = 0 \). Here, \( a = 4 \), \( b = -5 \), and \( c = k \). We will now use the condition \( \Delta = 0 \) to find \( k \).
Substitute the values of \( a \), \( b \), and \( c \) into the discriminant formula:
\[ \Delta = (-5)^2 - 4 \times 4 \times k \] Simplify the expression:
\[ \Delta = 25 - 16k \] Since the roots are real and equal, we set the discriminant equal to zero:
\[ 25 - 16k = 0 \]

Step 3: Solve for \( k \):
Solving the equation \( 25 - 16k = 0 \), we get:
\[ 16k = 25 \] \[ k = \frac{25}{16} \]

Step 4: Conclusion:
The value of \( k \) for which the roots of the quadratic equation \( 4x^2 - 5x + k = 0 \) are real and equal is \( \frac{25}{16} \).