If $ a - b = 3 $ and $ a^2 + b^2 = 29 $, find the value of $ ab $.

### Concept & Formula This problem requires connecting three algebraic components—the difference of two numbers, the sum of their squares, and their product—using a single standard identity. The formula that links these three variables together is: $$ (a - b)^2 = a^2 + b^2 - 2ab $$ ### Step-by-Step Solution We are given two pieces of explicit information: 1. $ a - b = 3 $ 2. $ a^2 + b^2 = 29 $ We need to find the value of the product, $ab$. Start by writing out the relevant algebraic identity: $$ (a - b)^2 = a^2 + b^2 - 2ab $$ Substitute the known values into this equation: $$ (3)^2 = 29 - 2ab $$ Simplify the squared term on the left side: $$ 9 = 29 - 2ab $$ Rearrange the equation to isolate the term containing our unknown ($2ab$): Move $-2ab$ to the left side and $9$ to the right side: $$ 2ab = 29 - 9 $$ $$ 2ab = 20 $$ Divide by 2 to solve for $ab$: $$ ab = \frac{20}{2} $$ $$ ab = 10 $$ ### Exam Strategy & Shortcut For small numbers, you can sometimes guess the values of $a$ and $b$ using factors. We need two numbers with a difference of 3 and a sum of squares equal to 29. Quickly mentally scan small integers: $5$ and $2$. * $5 - 2 = 3$ (Matches) * $5^2 + 2^2 = 25 + 4 = 29$ (Matches) Therefore, $a=5$, $b=2$. Their product is $5 \times 2 = 10$. This mental "guess and check" is often faster than the formula for very small whole numbers. ### Common Pitfall A common error is confusing the formula with $ (a+b)^2 $ and using $ +2ab $ instead of $ -2ab $, which results in $ 9 = 29 + 2ab $, leading to a negative (and incorrect) product of $-10$. Always verify the signs in your algebraic expansions. ### Final Answer **Therefore, the correct answer is 10.**