The average of a non-zero real number and its square is equal to 11 times the number itself. What is the value of this number?

Introduction / Context:
This is an algebraic average problem involving a number and its square. We are told that the average of a non-zero number and its square is eleven times the number itself. We must translate this statement into an equation in one variable and then solve the resulting quadratic equation to find the number.

Given Data / Assumptions:

• Let the non-zero number be x.
• The average of x and x^2 is 11x.
• We are told x is not zero.
• We must determine the value of x from the given options.

Concept / Approach:
The average of two quantities is the sum of the quantities divided by 2. So the average of x and x^2 is: (x + x^2) / 2. We are told this equals 11x, so we can set up the equation: (x + x^2) / 2 = 11x. Multiplying both sides by 2 eliminates the fraction, after which we obtain a standard quadratic equation that can be factored.

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 clear the denominator. x + x^2 = 22x. Step 3: Rearrange the equation to one side to form a quadratic. x^2 + x - 22x = 0 → x^2 - 21x = 0. Step 4: Factor the quadratic expression. x^2 - 21x = x(x - 21) = 0. Step 5: Set each factor equal to zero. x = 0 or x = 21. Step 6: The problem explicitly states that the number is non-zero, so we discard x = 0. Thus, x = 21 is the required number.
Verification / Alternative check:
Check x = 21 in the original condition. Number = 21, square = 21^2 = 441. Average = (21 + 441) / 2 = 462 / 2 = 231. 11 times the number = 11 * 21 = 231. Both sides match, confirming that 21 satisfies the condition.
Why Other Options Are Wrong:
Substituting 22, 11, 17 or 14 into the equation (x + x^2) / 2 = 11x will not satisfy it; the left and right sides become unequal. For example, if x = 22, average = (22 + 484) / 2 = 506 / 2 = 253, while 11 * 22 = 242, which does not match.
Common Pitfalls:
A common mistake is to forget to clear the fraction properly or to mishandle the movement of terms when forming the quadratic equation. Some students might also forget the condition that x is non-zero and mistakenly keep x = 0 as a valid solution.
Final Answer:
The non-zero number that satisfies the condition is 21.