$\sqrt{x} \times \sqrt{484} = 1034$

### Concept & Strategy Isolate the algebraic term $\sqrt{x}$ by evaluating the known perfect square root term and dividing it through to the right-hand side. Core algebraic identity: $$ \text{If } \sqrt{x} \times a = b, \text{ then } \sqrt{x} = \frac{b}{a} $$ ### Step-by-Step Solution - **Step 1: Extract the known square root.** - Recognize that $\sqrt{484} = 22$. - **Step 2: Substitute and simplify.** - $\sqrt{x} \times 22 = 1034$ - $\sqrt{x} = \frac{1034}{22}$ - Divide $1034$ by $22$: $\sqrt{x} = 47$ - **Step 3: Square both sides to find $x$.** - $x = 47^2$ - $47^2 = (50 - 3)^2 = 2500 - 300 + 9 = 2209$ ### Exam Strategy & Shortcut After finding $\sqrt{x} = 47$, determine $47^2$ quickly by referencing $50^2 = 2500$. Since $47$ is just below $50$, its square must be somewhat less than $2500$. Furthermore, $7^2 = 49$, so the final answer must end in $9$. Both $2209$ and $2401$ fit this, but $2401$ is $49^2$, making $2209$ the correct choice. ### Common Pitfall Mistakenly taking the square root of $47$ instead of squaring it, or halting calculations once you discover $47$ because of an operational mixup. Always remember that removing a square root demands squaring the opposing balance. ### Final Answer Therefore, the correct answer is 2209.