Evaluate x²+9/x² Given x+3/x=5: No Solving Required

We want to calculate the value of the expression:

$x^2+\frac{9}{x^2}=\text{?}$

based on the given equation:

$x+\frac{3}{x}=5$

but without solving it for x,

For this, let's first notice that while the given deals with terms with first power only,

in the expression we want to calculate - there are terms with second power only,

therefore we understand that apparently we need to square the expression on the left side of the given equation,

We'll remember of course the shortened multiplication formula for binomial square:

$(a\pm b)^2=a^2\pm2ab+b^2$

and we'll square both sides of the given equation, later we'll emphasize something worth noting that happens in the given mathematical structure in the expression in question (the mathematical structure where a term and its proportional inverse are added):

$x+\frac{3}{x}=5 \hspace{6pt}\text{/}()^2\\ (x+\frac{3}{x})^2=5^2\\ \downarrow\\ x^2+2\cdot \textcolor{blue}{x\cdot \frac{3}{x}}+ \frac{3^2}{x^2}=25\\ \downarrow\\ x^2+2\cdot \textcolor{blue}{3}+ \frac{9}{x^2}=25\\$Let's now notice that the "mixed" term in the shortened multiplication formula ($2ab$) gives us - from squaring the mathematical structure in question - a free number, meaning - it's not dependent on the variable x, since it involves multiplication between an expression with a variable and its proportional inverse,

This fact actually allows us to isolate the desired expression from the equation we get and find its value (which is not dependent on the variable) even without knowing the value of the unknown (or unknowns) that solves the equation:

$x^2+2\cdot \textcolor{blue}{3}+ \frac{9}{x^2}=25\\ x^2+6+ \frac{9}{x^2}=25\\ \boxed{x^2+\frac{9}{x^2}=19}$

Therefore the correct answer is answer C.