Twenty times a positive integer is less than its square by 96. What is the integer?

### Concept & Formula This word problem constructs a quadratic equation by relating a number to its square and a constant difference. $$ 20x = x^2 - 96 $$ ### Step-by-Step Solution * Let the positive integer be $x$. * Twenty times the integer is $20x$. * The problem states this value is 96 less than its square ($x^2$): $$ 20x = x^2 - 96 $$ * Rearrange the equation into a standard quadratic form ($ax^2 + bx + c = 0$): $$ x^2 - 20x - 96 = 0 $$ * We need two numbers that multiply to -96 and add up to -20. Factors of 96 include: $1 \times 96$, $2 \times 48$, $3 \times 32$, $4 \times 24$, $6 \times 16$, $8 \times 12$. The pair $-24$ and $+4$ satisfies both conditions: $(-24) \times 4 = -96$ and $-24 + 4 = -20$. * Factor the quadratic: $$ x^2 - 24x + 4x - 96 = 0 $$ $$ x(x - 24) + 4(x - 24) = 0 $$ $$ (x - 24)(x + 4) = 0 $$ * The possible solutions are $x = 24$ and $x = -4$. * Since the problem specifies a *positive* integer, we disregard -4. ### Exam Strategy & Shortcut Use the **Option Elimination Strategy** to bypass factoring entirely. * Test (a) 20: $20 \times 20 = 400$. Is 400 less than $20^2$ (400) by 96? No, they are equal. * Test (b) 24: $24 \times 20 = 480$. Square of 24 is 576. Difference: $576 - 480 = 96$. This perfectly satisfies the condition. Plugging in the answers takes only seconds compared to solving a quadratic equation. ### Common Pitfall When writing the initial equation, a common mistake is translating "less than its square by 96" as $96 - x^2$ instead of $x^2 - 96$. Always subtract the specific amount *from* the base value. ### Final Answer **Therefore, the correct answer is 24.**