Calculate x² + 1/x² Given x + 1/x = 4: Indirect Solution Method

We want to calculate the value of the expression:

$x^2+\frac{1}{x^2}=\text{?}$

based on the given equation:

$x+\frac{1}{x}=4$

but without solving it for x,

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

in the expression whose value 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 a 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 inverse are added):

$x+\frac{1}{x}=4 \hspace{6pt}\text{/}()^2\\ (x+\frac{1}{x})^2=4^2\\ \downarrow\\ x^2+2\cdot \textcolor{blue}{x\cdot \frac{1}{x}}+ \frac{1}{x^2}=16\\ \downarrow\\ x^2+2\cdot \textcolor{blue}{1}+ \frac{1}{x^2}=16\\$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's a multiplication between an expression and its 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:

$x^2+2\cdot \textcolor{blue}{1}+ \frac{1}{x^2}=16\\ x^2+2+ \frac{1}{x^2}=16\\ \boxed{x^2+\frac{1}{x^2}=14}$

Therefore the correct answer is answer D.