If the sum of a number and its square is 182, what is the number?

### Concept & Formula This problem requires establishing a quadratic equation since it involves the addition of a variable and its squared self. $$ x + x^2 = 182 $$ ### Step-by-Step Solution * Let the unknown number be $x$. * The sum of the number and its square is given as 182: $$ x + x^2 = 182 $$ * Rearrange into standard quadratic form ($ax^2 + bx + c = 0$): $$ x^2 + x - 182 = 0 $$ * We need to find two numbers that multiply to -182 and add to 1. Testing near squares: $13^2 = 169$ and $14^2 = 196$. $14 \times (-13) = -182$ and $14 + (-13) = 1$. * Factor the quadratic equation: $$ x^2 + 14x - 13x - 182 = 0 $$ $$ x(x + 14) - 13(x + 14) = 0 $$ $$ (x - 13)(x + 14) = 0 $$ * The possible solutions are $x = 13$ or $x = -14$. * Now look at the options provided: (a) 15, (b) 26, (c) 28, (d) 91. Neither 13 nor -14 is listed. ### Exam Strategy & Shortcut Instead of factoring a large quadratic, use **Option Elimination** and mental math based on squares. * $15^2 = 225$, which is already larger than 182. (Eliminated) * 26, 28, and 91 are much larger. Their squares will be massive. (Eliminated) Since all given numerical options are trivially too large, you can instantly conclude the answer must be "None of these" without doing any deep calculation. ### Common Pitfall Students sometimes overcomplicate their factoring or waste time trying to force one of the given options (like 15) to work. If simple estimations show the numbers are too big, trust your intuition and select the "None of these" trapdoor. ### Final Answer **Therefore, the correct answer is None of these.**