Question

Under what conditions is the expression$\frac{a^2+ab+b^2}{a^2-b^2}$undefined.

Short answer

The expression is undefined for $a=-b,b$.

Step-by-step solution

Step 1. Write down the given information.

The given expression is$\frac{a^2+ab+b^2}{a^2-b^2}$

Step 2. Concept used.

The expression will be undefined for which the denominator of the given expression will be equal to zero.

Step 3. Calculation.

The given expression $\frac{a^2+ab+b^2}{a^2-b^2}$has denominator $a^2-b^2$. Therefore,

equating $a^2-b^2=0$gives,

role="math" localid="1648315877907" $\begin{array}{c}{a}^2-b^2=0\\ \left(a-b\right)\left(a+b\right)=0\\ a-b=0\text{\hspace{0.33em}or}a+b=0\\ a=b\text{\hspace{0.33em}or}a=-b\\ a=\left\{-b,b\right\}\\ a=-b,b\end{array}$

Since the expression is undefined for $a=-b,b$.

Step 4. Conclusion.

The expression is undefined for $a=-b,b$.