Expression Evaluation: Calculate a²x² - 6ax + 14 Given ax - 3 = 1

 In order to calculate the value of the expression in the problem:

$a^2x^2-6ax+14$

Based on the given that:

$ax-3=1$

and without solving an equation (meaning - without finding the value of x, in this case),

we will use the method of completing the square,

First, let's recall the principles of this method and its general idea, and demonstrate it with a simpler expression:

In this method, we use the perfect square formulas in order to give the expression the form of a perfect square,

This method is called "completing the square" because in this method we "complete" a missing part to a certain expression in order to get from it the form of a perfect square,

That is, we use the formulas for perfect squares:

$(c\pm d)^2=c^2\pm2cd+d^2$

and we'll bring the expression to a square form by adding and subtracting the missing term,

For example:

Let's represent the expression:

$x^2-4x+3$

as a perfect square expression plus a correction,

First, we'll try to give the current expression a form that resembles the right side of the perfect square formulas mentioned, we'll also identify that we're interested in the subtraction form of the perfect square formula, since the non-squared term in the given expression has

a negative sign, we'll continue,

First, let's deal with the two terms with the highest powers in the expression:

$\underline{x^2-4x}+3 \textcolor{blue}{\leftrightarrow} \underline{ c^2-2cd+d^2 }\\ \downarrow\\ \underline{x^2-2\cdot x\cdot 2}+3 \textcolor{blue}{\leftrightarrow} \underline{ c^2-2cd+d^2} \\$We identify that if we want to get from these two terms (underlined below in the calculation) a perfect square form,

We'll need to add to these two terms the term$2^2$, but we don't want to change the value of the expression in question, so we'll also subtract this number from the expression,

That is, we'll add and subtract the number (or expression) we need to "complete" to a perfect square form,

In the following calculation, the "trick" is demonstrated (two lines under the term we added and subtracted),

Later - we'll put it in perfect square form (demonstrated with colors) and in the final stage we'll further simplify the expression:

$x^2-2\cdot x\cdot 2+3\\ x^2-2\cdot x\cdot 2\underline{\underline{+2^2-2^2}}+3\\ \textcolor{red}{x}^2-2\cdot \textcolor{red}{x}\cdot \textcolor{green}{2}+\textcolor{green}{2}^2-4+3\\ \downarrow\\ (\textcolor{red}{x}-\textcolor{green}{2})^2-4+3\\ \downarrow\\ \boxed{(x-2)^2-1}$

Therefore- we got the completing the square form for the given expression,

We won't expand here but we'll note that there are several different completion forms, since we can "rotate" the perfect square formulas in different ways,

Let's return then to the problem and examine again the expression on which we want to perform "completing the square":

$a^2x^2-6ax+14$We'll also remember that we want to use the information that:

$ax-3=1$Therefore, first we'll present the first term (which is the squared term in the expression) as a perfect square, and we'll examine the first two terms in comparison to the perfect square formula while we identify the expression form in the perfect square formula and try to understand what is the missing term in order to perform the completing the square:

$\underline{ a^2x^2-6ax} +14 \textcolor{blue}{\leftrightarrow} \underline{ c^2-2cd+d^2 }\\ \underline{ (ax)^2-6ax} +14 \textcolor{blue}{\leftrightarrow} \underline{ c^2-2cd+d^2 }\\ \downarrow\\ \underline{ (\textcolor{green}{ax})^2\textcolor{red}{-2}\cdot \textcolor{green}{ax}\cdot\textcolor{purple}{3}} +14 \textcolor{blue}{\leftrightarrow} \underline{ c^2\textcolor{red}{-2}\textcolor{green}{c}\textcolor{purple}{d}\hspace{2pt}\boxed{+d^2} }\\$Again, we identify that in order to complete the expression of the first two terms to a square form, we need to add and subtract the number $3^2$ , we'll do this, then we'll get the square form using the perfect square formula and simplify the resulting expression:

$(ax)^2-2\cdot ax\cdot 3+14\\ (ax)^2-2\cdot ax\cdot 3\underline{\underline{+3^2-3^2}}+14\\ (\textcolor{red}{ax})^2-2\cdot \textcolor{red}{ax}\cdot \textcolor{green}{2}+\textcolor{green}{3}^2-9+14\\ \downarrow\\ (\textcolor{red}{ax}-\textcolor{green}{3})^2-9+14\\ \downarrow\\ \boxed{(ax-3)^2+5}$

To summarize, we got therefore that:

$a^2x^2-6ax +14=\\ (ax)^2-2\cdot ax\cdot3+14=\\ (ax-3)^2-9+14=\\ \boxed{(ax-3)^2}+5$

Now we'll use the given that:

$ax-3=1$

We'll use this given "completely" and substitute its value in the expression we got:

$a^2x^2-6ax +14=\\ (\underline{ax-3})^2+5 =\text{?}\leftrightarrow \underline{ax-3}=1\\ \downarrow \\ \underline{1}^2+5=\\ \boxed{6}$

Therefore the correct answer is answer C.