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

Our goal is to calculate the value of the following expression:

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

based on the given equation:

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

However without solving it for x,

Note that while the given equation deals with terms raised to the first power only,

in the expression we want to calculate - there are terms raised to the second power.

Therefore we need to square the expression on the left side of the given equation.

We can do this by using the shortened multiplication formula for binomial square:

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

We'll proceed to square both sides of the given equation, later on we'll discuss what happens in the given mathematical structure when a term and its proportional inverse are added:

$x+\frac{3}{x}=5 \hspace{6pt}\text{/}()^2\\ (x+\frac{3}{x})^2=5^2\\ \downarrow\\ x^2+2\cdot \textcolor{blue}{x\cdot \frac{3}{x}}+ \frac{3^2}{x^2}=25\\ \downarrow\\ x^2+2\cdot \textcolor{blue}{3}+ \frac{9}{x^2}=25\\$For now take note that the "mixed" term in the shortened multiplication formula ($2ab$) gives us - as a result of squaring the mathematical structure in question - a free number. This signifies that 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 obtain. We can subsequently determine 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.