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

To determine the value of \(m\) for which the quadratic equation \(3x^2 - 7x + m = 0\) has real and equal roots, we need to use the concept of the discriminant of a quadratic equation. The discriminant \(\Delta\) is given by the formula:

\(\Delta = b^2 - 4ac\)

where \(a, b,\) and \(c\) are the coefficients of the quadratic equation \(ax^2 + bx + c = 0\).

Here, the given quadratic equation is \(3x^2 - 7x + m = 0\), so we have:

• \(a = 3\)
• \(b = -7\)
• \(c = m\)

For real and equal roots, the discriminant must be zero:

\(\Delta = 0\)

Substitute the values into the discriminant formula:

\((-7)^2 - 4 \cdot 3 \cdot m = 0\)

This simplifies to:

\(49 - 12m = 0\)

Solving for \(m\):

1. Rearrange the equation to isolate \(m\):
2. \(12m = 49\)
3. Divide both sides by 12:
4. \(m = \frac{49}{12}\)

Thus, the value of \(m\) for which the quadratic equation has real and equal roots is \(\frac{49}{12}\).

The correct option is \(\frac{49}{12}\).

Let's verify by substituting back:

If \(m = \frac{49}{12}\), the quadratic equation becomes:

\(3x^2 - 7x + \frac{49}{12} = 0\)

Calculating the discriminant again:

\(\Delta = (-7)^2 - 4 \cdot 3 \cdot \frac{49}{12}\)

This confirms:

\(49 - \frac{147}{12} = 49 - 12.25 = 0\)

Therefore, the calculation is correct and supports our answer.