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:

For any quadratic equation $ax^2 + bx + c = 0$ to have real and equal roots, remember the direct condition $b^2 = 4ac$.
This allows you to quickly set up the equation and solve for the unknown parameter without extra intermediate steps.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given a quadratic equation $3x^2 - 7x + m = 0$.
We need to find the value of the constant $m$ such that the quadratic equation has real and equal roots.
The nature of the roots of any quadratic equation is determined by its discriminant.

Step 2: Key Formula or Approach:
For a standard quadratic equation $ax^2 + bx + c = 0$, the roots are real and equal if and only if the discriminant $D$ is equal to zero:
\[ D = b^2 - 4ac = 0 \]
We identify the coefficients $a$, $b$, and $c$ from the given equation and solve for $m$.

Step 3: Detailed Explanation:

• Identify the coefficients of the given quadratic equation $3x^2 - 7x + m = 0$:
\[ a = 3, \quad b = -7, \quad c = m \]

• Set the discriminant $D$ to zero for real and equal roots:
\[ b^2 - 4ac = 0 \]

• Substitute the values of $a$, $b$, and $c$:
\[ (-7)^2 - 4(3)(m) = 0 \]

• Simplify the equation:
\[ 49 - 12m = 0 \]

• Isolate the variable $m$:
\[ 12m = 49 \]
\[ m = \frac{49}{12} \]


Step 4: Final Answer:
The value of $m$ for which the equation has real and equal roots is $\frac{49}{12}$.