Question 176·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

Type $x+y=14$ and $xy=45$ on separate Desmos lines. An intersection lies on both graphs, so its coordinates satisfy both equations. Click the two intersections: $(5,9)$ and $(9,5)$. The pair changes order, but the two numbers are the same.

The absolute value $|x-y|$ is the nonnegative distance between the numbers. Swapping the numbers keeps that distance the same. Type $|5-9|$ on the next Desmos line; it shows $4$. So the two numbers differ by $4$. Choice B.

You can find the gap without finding either number. Squaring the sum creates a $+2xy$ term. Subtracting $4xy$ changes it to $-2xy$, which makes the squared difference. Expand and subtract:

The problem gives $x+y=14$ and $xy=45$, so replace each whole quantity in that identity:

The square root of $(x-y)^2$ is $|x-y|$, not necessarily $x-y$, because a square root is nonnegative. Type $\sqrt{14^2-4(45)}$ in Desmos; it shows $4$. So the absolute difference is $4$. Choice B.