1959 AHSME Problems/Problem 35

Problem 35

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: $\textbf{(A)}\ x\ge 0 \qquad\textbf{(B)}\ x\le n\qquad\textbf{(C)}\ x=0\qquad\textbf{(D)}\ \text{the set of all real numbers}\qquad\textbf{(E)}\ \text{none of these}$

Applying the difference of squares technique on this problem, we can see that $$(x-m)^2-(x-n)^2=((x-m)+(x-n)) \cdot ((x-m)-(x-n)),$$ so $$(x-m)^2-(x-n)^2=((x-m)+(x-n)) \cdot ((x-m)-(x-n))=(m-n)^2.$$ Simplifying gives us$$(2x-m-n)\cdot(n-m)=(m-n)^2.$$Negating $n-m$ creates:$$-(2x-m-n)\cdot(m-n)=(m-n)^2.$$Dividing by $m-n$, $$-(2x-m-n)=m-n$$ $$-2x+m+n=m-n$$ $$-2x+n=-n$$ $$-2x=-2n$$ $$x=n$$Lastly, since $n$ is a fixed negative number, $x$ must also be a fixed negative number, so $x<0$. Since this answer is not $A, B, C,$ or $D$, the solution must be $(E)$ none of these.