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The given image contains a mathematical expression in Gujarati. Here is the problem, translated and broken down for better understanding:

The problem presents a quadratic polynomial \( p(x) = 6x^2 - 13x + 3m - 9i \) (where \( i \) likely represents the imaginary unit, i.e., \( i = \sqrt{-1} \)).

The question asks: If the roots of the polynomial are real and equal, find the value of \( m \).

### Solution:
For a quadratic equation \( ax^2 + bx + c = 0 \), the condition for the roots to be real and equal is that the discriminant must be zero. The discriminant \( \Delta \) for a quadratic equation \( ax^2 + bx + c \) is given by:

\[
\Delta = b^2 - 4ac
\]

For the quadratic equation \( p(x) = 6x^2 - 13x + (3m - 9i) \), comparing it with the general quadratic form \( ax^2 + bx + c \):

- \( a = 6 \)
- \( b = -13 \)
- \( c = 3m - 9i \)

To satisfy the condition for equal roots, the discriminant must be zero:

\[
\Delta = (-13)^2 - 4(6)(3m - 9i) = 0
\]

This simplifies to:

\[
169 - 24(3m - 9i) = 0
\]

\[
169 - 72m + 216i = 0
\]

Equating the real and imaginary parts separately, we get two equations:

1. For the real part:
   \[
   169 - 72m = 0 \quad \Rightarrow \quad m = \frac{169}{72}
   \]

2. For the imaginary part:
   \[
   216i = 0 \quad \Rightarrow \quad i = 0
   \]

Thus, the value of \( m \) is \( \frac{169}{72} \).

Would you like further clarification on any part of the solution, or do you have any other questions?

Here are 5 related questions to explore further:
1. How do you solve a quadratic equation when the roots are not equal?
2. What is the significance of the discriminant in a quadratic equation?
3. How do you handle complex roots in a quadratic equation?
4. Can the discriminant be negative? If so, what does that indicate about the roots?
5. How does the value of \( m \) change if the polynomial has real but unequal roots?

**Tip:** Always check whether the discriminant of a quadratic equation is positive, zero, or negative. This determines whether the roots are real, equal, or complex.

જો વિધાત બહુપદી p(x) = 6x² - 13x + 3m - 9i શુન્ય પરસપર વ્યસ્ત હોય, તો m = .........

Explore how to solve the quadratic polynomial p(x) = 6x² - 13x + 3m - 9i to find the value of m when the roots are real and equal. This solution includes key concepts such as the discriminant and the condition for equal roots, appropriate for students in Grades 10-12.

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