X^2 - 9x + 16 = 0

To solve the quadratic equation \(x^2 - 9x + 16 = 0\), we will first find the roots \(x_1\) and \(x_2\) using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \(a = 1\), \(b = -9\), and \(c = 16\).

Calculate the discriminant:

\[
b^2 - 4ac = (-9)^2 - 4 \times 1 \times 16 = 81 - 64 = 17
\]

Now apply the quadratic formula:

\[
x_{1,2} = \frac{9 \pm \sqrt{17}}{2}
\]

Thus, the roots are:

\[
x_1 = \frac{9 + \sqrt{17}}{2}, \quad x_2 = \frac{9 - \sqrt{17}}{2}
\]

Next, we need to calculate the value of the expression:

\[
T = \frac{x_1 \sqrt{x_2} + x_2 \sqrt{x_1}}{x_1^2 + x_2^2}
\]

First, let's find \(x_1^2 + x_2^2\):

\[
x_1 + x_2 = 9, \quad \text{and} \quad x_1 x_2 = 16
\]

Using the identity:

\[
x_1^2 + x_2^2 = (x_1 + x_2)^2 - 2x_1 x_2
\]

We find:

\[
x_1^2 + x_2^2 = 9^2 - 2 \times 16 = 81 - 32 = 49
\]

Next, we find \(x_1 \sqrt{x_2} + x_2 \sqrt{x_1}\):

Let \( \sqrt{x_1} = a \) and \( \sqrt{x_2} = b \), thus:

\[
x_1 = a^2, \quad x_2 = b^2
\]

Then:

\[
a^2 b + b^2 a = ab(a + b)
\]

Now, \(a^2 + b^2 = 9\) and \(ab = \sqrt{x_1 x_2} = 4\). Also, \(a+b\) can be found using:

\[
a^2 + b^2 = (a + b)^2 - 2ab
\]

Let \(s = a + b\):

\[
9 = s^2 - 8 \implies s^2 = 17 \implies s = \sqrt{17}
\]

Thus:

\[
\sqrt{x_1} + \sqrt{x_2} = \sqrt{17}
\]

Now combining everything for \(T\):

\[
T = \frac{4\sqrt{17}}{49}
\]

Hence, the value of \(T\) is:

\[
T = \frac{4\sqrt{17}}{49}
\]