Question 39·Easy·Advanced Math

To graph the given expression in Desmos, use $x$ in place of $z$ and type $y=\frac{4x^2-25}{2x+5}$. Desmos draws what looks like a straight line. The given $z\ne-\frac52$ means one input is missing, even if you can’t see that gap on the graph. Factoring will explain the line.

The numerator has two squares with a minus between them: $4z^2$ is the square of $2z$, and $25$ is the square of $5$. Rewrite it to expose that pattern:

A difference of squares follows $a^2-b^2=(a-b)(a+b)$. It works because the two middle terms cancel when you multiply the parentheses back out. Apply it with $a=2z$ and $b=5$, keeping the factors in symbols for now:

Substitute $a=2z$ and $b=5$ into the factors:

The denominator is the whole factor $2z+5$. The problem excludes $z=-\frac52$, where that factor would be $0$, so you can divide the top and bottom by it for every allowed $z$. Cancel that factor:

Cancel whole factors, not pieces of sums. The equivalent expression is $2z-5$. Choice D.