Question

The value of \(m\) for which the quadratic equation \(3x^2 - 7x + m = 0\) has real and equal roots, is

• A. \(7\)
• B. \(\frac{49}{12}\)
• C. \(\frac{49}{3}\)
• D. \(4\)

Hint:

"Real and equal roots" \(\rightarrow D=0\).
"Real roots" \(\rightarrow D \ge 0\).
Make sure you distinguish between these two phrasing in exams.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A quadratic equation \(ax^2 + bx + c = 0\) has real and equal roots if and only if its discriminant (\(D\)) is zero.
Step 2: Key Formula or Approach:
The discriminant is given by:
\[ D = b^2 - 4ac = 0 \]
Step 3: Detailed Explanation:
For the equation \(3x^2 - 7x + m = 0\):
\(a = 3, b = -7, c = m\)
Set \(D = 0\):
\[ (-7)^2 - 4(3)(m) = 0 \]
\[ 49 - 12m = 0 \]
\[ 12m = 49 \]
\[ m = \frac{49}{12} \]
Step 4: Final Answer:
The value of \(m\) is \(\frac{49}{12}\).