Let's go through the questions visible in the image:

Question 14:

Write the roots of the quadratic equation $x(x - 6) = 0$.

To solve $x(x - 6) = 0$, we set each factor equal to zero:

1. $x = 0$
2. $x - 6 = 0$ → $x = 6$

The roots of the equation are 0 and 6.

Question 15:

Express 75 as a product of its prime factors.

We can find the prime factorization of 75 as follows:

1. $75 \div 3 = 25$
2. $25 \div 5 = 5$
3. $5 \div 5 = 1$

Thus, $75 = 3 \times 5^2$.

Question 16:

If 5, $x$, and 11 are in arithmetic progression, find the value of $x$.

In an arithmetic progression, the difference between consecutive terms is constant. So, the difference between $5$ and $x$ should be equal to the difference between $x$ and $11$.

Set up the equation: $x - 5 = 11 - x$ Solve for $x$: $x - 5 = 11 - x$ $2x = 16$ $x = 8$

The value of $x$ is 8.

Would you like further details or have any questions? Here are some related questions to deepen understanding:

1. How do you factorize more complex quadratic equations?
2. Can you explain how to verify the prime factorization of a number?
3. What is the general form of an arithmetic progression?
4. Can a quadratic equation have more than two roots?
5. What are the properties of numbers in an arithmetic progression?

Tip: When solving quadratic equations, always check if it can be factored before using the quadratic formula for simplicity.