https://file.aiquickdraw.com/mathbot/file/17273718636230urwygl6.jpg

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.

14. Write the roots of the quadratic equation x(x - 6) = 0. 15. Express 75 as a product of its prime factors. 16. If 5, x, and 11 are in arithmetic progression, then find the value of x.

Learn how to solve quadratic equations, express numbers as prime factors, and find terms in arithmetic progression. This set of problems covers algebraic concepts including quadratic equations, prime factorization, and arithmetic progressions, suitable for students in Grades 8-9.

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