Question 233·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

Choose $a=3$ and $b=2$. In Desmos, use $x$ in place of $c$ and type $y=\frac{a}{b-x}$ and $y=\frac{2b}{x-a}$. Click where the graphs meet: Desmos shows about $(2.429,-7)$, so $c\approx2.429$ for this sample. That value is neither $2$ nor $3$, so both original denominators are defined.

Keep $a=3$ and $b=2$, then type the first three proposed expressions. Desmos gives $1$, about $0.143$, and about $0.412$, in order. None matches the $c$-value near $2.429$ from the original equation, so none of those formulas works for every allowed $a$ and $b$.

Type the remaining expression, $\frac{2b^2+a^2}{a+2b}$. Desmos gives about $2.429$, matching the intersection. Since every other choice failed this allowed sample, the equation expressing $c$ is $c=\frac{2b^2+a^2}{a+2b}$. Choice D.

No Desmos needed. Rearranging the letters proves the formula for every allowed value. The problem says $b\ne c$ and $c\ne a$, so neither denominator is zero. Multiply both whole sides by $(b-c)(c-a)$:

Distribute $a$ and $2b$, keeping the minus signs inside the parentheses:

Add $2bc$ to both sides so both terms containing $c$ are on the left:

Add $a^2$ to both sides. Moving $-a^2$ makes it positive on the right:

Both terms on the left contain $c$, so factor it out: $c(a+2b)=2b^2+a^2$. When the target letter appears twice, gather its terms and factor it out before dividing.

The problem gives $a+2b\ne0$, so divide both sides by $a+2b$: $c=\frac{2b^2+a^2}{a+2b}$. This expresses $c$ in terms of $a$ and $b$. Choice D.