Question

The symbol $\ge$ means “greater than or equal to”; the symbol $\le$ means “less than or equal to.” In the equation $(x-m)^2-(x-n)^2=(m-n)^2,$ $m$ is a fixed positive number, and $n$ is a fixed negative number. The set of values $x$ satisfying the equation is:

$x\ge0$

$x\le n$

$x=0$

the set of all real numbers

none of these

Answer: E

Concepts:difference of squares linear equation inequality

Difficulty rating: 1280

Small Hint:

Factor the difference of squares on the left

Big Hint:

Because $m\ne n,$ divide by $m-n$ and solve for $x$

Solution:

Factoring the left side gives $\begin{aligned} &(n-m)(2x-m-n)\\ &\qquad=(m-n)^2. \end{aligned}$ Since $m\ne n,$ division by $n-m$ yields $2x-m-n=n-m,$ so $x=n.$ This single negative value is not any of choices A through D.

Therefore, the correct answer is E.

1959 AMC 12 Problem 35

Answer: E

Difficulty rating: 1280

Small Hint:

Big Hint:

Solution:

Problem 35 in Other Years