The average of a non-zero number and its square is 5 times the number. The number is

### Concept & Logic This problem requires translating a word problem directly into an algebraic equation. The core concept is equating the standard average formula to the condition given in the text. $$ \text{Average} = \frac{\text{Sum of Terms}}{\text{Number of Terms}} $$ ### Step-by-Step Solution * **Given:** * Let the non-zero number be $x$. * The two terms being averaged are $x$ and its square, $x^2$. * **Calculation:** * Set up the equation based on the given condition: * $\frac{x + x^2}{2} = 5x$ * Multiply both sides by $2$ to clear the denominator: * $x + x^2 = 10x$ * Rearrange into a standard quadratic equation: * $x^2 + x - 10x = 0$ * $x^2 - 9x = 0$ * Factor out $x$: * $x(x - 9) = 0$ * This gives two possible solutions: $x = 0$ or $x = 9$. * Since the problem specifies a "non-zero number", $x$ must be $9$. ### Exam Strategy & Shortcut You can solve this rapidly using Option Elimination. Test Option (a): Number is $9$. Square is $81$. Average $= \frac{9 + 81}{2} = \frac{90}{2} = 45$. Is $45$ equal to $5 \times 9$? Yes. The option is verified instantly without solving the quadratic equation. ### Common Pitfall Dividing both sides of $x + x^2 = 10x$ by $x$ without explicitly acknowledging that $x \neq 0$. While mathematically safe here due to the question's text, dividing by variables can lead to lost roots in standard algebra. Always factor instead. ### Final Answer **Therefore, the correct answer is 9.**