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.