Solve x² - 81 = 0: Finding Values in a Basic Quadratic Equation

Let's solve the given equation:

$x^2-81=0$Note that we can factor the expression on the left side using the difference of squares formula:

$(\textcolor{red}{a}+\textcolor{blue}{b}) (\textcolor{red}{a}-\textcolor{blue}{b})=\textcolor{red}{a}^2-\textcolor{blue}{b}^2$We'll do this using the fact that:

$81=9^2$Therefore, we'll represent the rightmost term as a squared term:

$x^2-81=0 \\ \downarrow\\ \textcolor{red}{x}^2-\textcolor{blue}{9}^2=0$If so, we can represent the expression on the left side in the above equation as a product of expressions:

$\textcolor{red}{x}^2-\textcolor{blue}{9}^2=0 \\ \downarrow\\ (\textcolor{red}{x}+\textcolor{blue}{9})(\textcolor{red}{x}-\textcolor{blue}{9})=0$From here we'll remember that the product of expressions equals 0 only if at least one of the multiplied expressions equals zero,

Therefore, we'll get two simple equations and solve them by isolating the variable in each:

$x+9=0\\ \boxed{x=-9}$or:

$x-9=0\\ \boxed{x=9}$

Let's summarize the solution of the equation:

$x^2-81=0 \\ \downarrow\\ \textcolor{red}{x}^2-\textcolor{blue}{9}^2=0 \\ \downarrow\\ (\textcolor{red}{x}+\textcolor{blue}{9})(\textcolor{red}{x}-\textcolor{blue}{9})=0 \\ x+9=0\rightarrow\boxed{x=-9}\\ x-9=0\rightarrow\boxed{x=9}\\ \downarrow\\ \boxed{x=9,-9}$Therefore, the correct answer is answer B.