For $f ( x ) = \sqrt { x } - \sqrt { x - 169 }$ , find x such that $f ( x ) = 1$ .
A) 14,450
B) 14,112
C) 28,900
D) 7,056
E) $7,225$

Question

Quizplus · Accepted Answer

To solve $f(x) = 1$, we set $\sqrt{x} - \sqrt{x - 169} = 1$. Squaring both sides gives $x - 2\sqrt{x(x - 169)} + x - 169 = 1$, simplifying to $2x - 170 = 2\sqrt{x(x - 169)}$. Dividing by 2 and squaring again yields $(x - 85)^2 = x(x - 169)$. Expanding and simplifying gives a quadratic equation in $x$, which, when solved, leads to $x = 7225$ as the solution that makes sense in the context of the original equation, since it must be greater than 169 to keep the square root of $x - 169$ real.

For f(x)=x−x−169f ( x ) = \sqrt { x } - \sqrt { x - 169 }f(x)=x​−x−169​

For $f ( x ) = \sqrt { x } - \sqrt { x - 169 }$