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

Problem:
We are given a quadratic equation:
\[ 4x^2 - 5x + k = 0 \]
We are told that the roots of this quadratic equation are real and equal, and we need to find the value of \(k\).

Step 1: Use the condition for equal roots
For a quadratic equation \( ax^2 + bx + c = 0 \), the condition for real and equal roots is that the discriminant must be zero:
\[ D = b^2 - 4ac = 0 \]

Step 2: Identify coefficients
From the equation \(4x^2 - 5x + k = 0\), we have:
\(a = 4\), \(b = -5\), \(c = k\)

Step 3: Apply the condition \(D = 0\)
\[ (-5)^2 - 4 \cdot 4 \cdot k = 0 \Rightarrow 25 - 16k = 0 \Rightarrow 16k = 25 \Rightarrow k = \frac{25}{16} \]

Final Answer:
The value of \(k\) for which the roots are real and equal is \(\frac{25}{16}\).