The average of a non zero number and its square is 11 times the number. What is the value of the number?

Introduction / Context:
This question combines the idea of average with a simple quadratic equation. You are given a relationship between a number, its square and a multiple of the number. Solving it correctly requires forming an equation and then using standard algebraic techniques to find the non zero solution.

Given Data / Assumptions:
- Let the number be x, where x is non zero.
- The average of x and x squared is 11 times the number x.
- Mathematically, this is expressed as (x + x^2) / 2 = 11x.
- We must find the value of x from the options 22, 21, 11 and 17.

Concept / Approach:
The average of two numbers is their sum divided by 2. So the statement directly becomes an equation in x. After simplifying, we get a quadratic equation. Quadratic equations can be solved by factoring or by using the quadratic formula. Here the equation factors easily, and one of the roots is zero, which we discard because the number is stated to be non zero.

Step-by-Step Solution:
Step 1: Write the equation for the average: (x + x^2) / 2 = 11x. Step 2: Multiply both sides by 2 to remove the denominator: x + x^2 = 22x. Step 3: Rearrange all terms to one side: x^2 + x - 22x = 0. Step 4: Simplify: x^2 - 21x = 0. Step 5: Factor the equation: x(x - 21) = 0. Step 6: The solutions are x = 0 or x = 21. Since the number is non zero, we take x = 21.

Verification / Alternative check:
Check x = 21 in the original condition. The average of 21 and 21^2 is (21 + 441) / 2 = 462 / 2 = 231. On the other side, 11 times 21 is 231. Both sides match, so x = 21 satisfies the condition.

Why Other Options Are Wrong:
- Option 22: For x = 22, (22 + 484) / 2 = 253, but 11 * 22 = 242, so the condition fails.
- Option 11: For x = 11, (11 + 121) / 2 = 66, but 11 * 11 = 121, not equal.
- Option 17: For x = 17, (17 + 289) / 2 = 153, but 11 * 17 = 187, again not equal.

Common Pitfalls:
Some students forget to multiply both sides by 2 correctly, or they move terms incorrectly, leading to sign errors. Another common mistake is forgetting to discard the zero root even when the question states the number is non zero. Always check which solutions are allowed by the wording of the question.

Final Answer:
The required non zero number is 21.