If the sum and difference of two numbers are 20 and 8 respectively, then the difference of their squares is

## Concept & Formula The difference of squares of two numbers can be directly calculated if you know their sum and their difference, by using the standard algebraic factorization identity. $$ x^2 - y^2 = (x + y)(x - y) $$ ## Step-by-Step Solution * **Given:** Let the two numbers be $x$ and $y$. Sum: $x + y = 20$ Difference: $x - y = 8$ * **Calculation / Deduction:** * We need to find the difference of their squares, which is $x^2 - y^2$. * According to the algebraic identity: * $$ x^2 - y^2 = (x + y)(x - y) $$ * Substitute the given values directly into the formula: * $x^2 - y^2 = 20 \times 8$ * $x^2 - y^2 = 160$ ## Exam Strategy & Shortcut **Direct Multiplication:** Recognize that "difference of squares" is simply the product of the sum and the difference. Instantly multiply $20 \times 8 = 160$. Do not waste time calculating the individual numbers ($14$ and $6$) just to square them and subtract ($196 - 36 = 160$). ## Common Pitfall A common inefficiency is solving for the individual numbers first. Adding the equations gives $2x = 28 \Rightarrow x = 14$, and subtracting gives $2y = 12 \Rightarrow y = 6$. While correct, squaring $14$ and $6$ takes more mental effort and time than simply multiplying the given sum and difference. ## Final Answer **Therefore, the correct answer is 160.**