$\sqrt{x} + 14 = \sqrt{2601}$

Concept & Logic To solve this equation, you must first evaluate the perfect square root on the right side of the equation. Once calculated, use basic algebra to isolate $\sqrt{x}$ and then square both sides to find $x$. $$ \text{If } \sqrt{x} = y, \text{ then } x = y^2 $$ Step-by-Step Solution * **Given:** $\sqrt{x} + 14 = \sqrt{2601}$ * **Calculation:** First, determine the square root of 2601. We know $50^2 = 2500$. Since 2601 ends in 1, its square root must end in 1 or 9. The number 2601 is very close to 2500, so the root is 51. $\sqrt{2601} = 51$ * Substitute this back into the equation: $\sqrt{x} + 14 = 51$ * Isolate $\sqrt{x}$ by subtracting 14 from both sides: $\sqrt{x} = 51 - 14$ $\sqrt{x} = 37$ * Square both sides to solve for $x$: $x = 37^2$ $x = 1369$ Exam Strategy & Shortcut Memorize squares up to 40. Knowing $37^2 = 1369$ instantly saves you from having to do manual multiplication during the final step. To quickly verify $37^2$, you can use the $(a-b)^2$ method: $(40-3)^2 = 1600 - 240 + 9 = 1369$. Common Pitfall A frequent mistake is stopping at $\sqrt{x} = 37$ and looking for 37 in the options (if it were present), or incorrectly doubling 37 instead of squaring it. Always remember to apply the inverse operation (squaring) to remove the radical. Final Answer **Therefore, the correct answer is 1369.**