Completing the Square: Find 3x-5 in 9x²-30x+4=0

 Let's first recall the principles of the "completing the square" method and its general idea:

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

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

That is, we use the formulas for perfect squares:

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

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

In the given problem let's first look at the given equation:

$9x^2-30x+4=0$

First, we'll try to give the expression on the left side of the equation a form that resembles the right side of the perfect square formulas mentioned above, we also identify that we are interested in the subtraction form of the perfect square formula, because the non-square term in the given expression, $$$30x$,

has a negative sign,

Let's continue,

First, let's deal with the two terms with the highest powers in the expression on the left side of the equation,

and we'll try to identify the missing term by comparing it to the perfect square formula,

To do this- first we'll present these terms in a form similar to the form of the first two terms in the perfect square formula:

$\underline{9x^2-30x}+4 \textcolor{blue}{\leftrightarrow} \underline{ c^2-2cd+d^2 }\\ \downarrow\\ \underline{(\textcolor{red}{3x})^2\stackrel{\downarrow}{-2 }\cdot \textcolor{red}{3x}\cdot \textcolor{green}{5}}+4 \textcolor{blue}{\leftrightarrow} \underline{ \textcolor{red}{c}^2\stackrel{\downarrow}{-2 }\textcolor{red}{c}\textcolor{green}{d}\hspace{2pt}\boxed{+\textcolor{green}{d}^2}} \\$We can notice that compared to the perfect square formula (which is on the right side of the blue arrow in the previous calculation) we are actually making the analogy:

$\begin{cases} 3x\textcolor{blue}{\leftrightarrow}c\\ 5\textcolor{blue}{\leftrightarrow}d \end{cases}$

Therefore, we identify that if we want to get a perfect square form from these two terms (underlined in the calculation),

We will need to add to these two terms the term$5^2$, but we don't want to change the value of the expression in question, so we'll also subtract this term from the expression,

In other words- we'll add and subtract the term (or expression) we need to "complete" to a perfect square form,

The next calculation demonstrates the "trick" (two lines under the term we added and subtracted from the expression),

Next- we'll put into perfect square form the appropriate expression (demonstrated using colors) and in the final stage we'll simplify the expression further:

$(3x)^2-2\cdot 3x\cdot 5+4\\ (3x)^2-2\cdot 3x\cdot 5\underline{\underline{+5^2-5^2}}+4\\ (\textcolor{red}{3x})^2-2\cdot \textcolor{red}{3x}\cdot \textcolor{green}{5}+\textcolor{green}{5}^2-25+4\\ \downarrow\\ (\textcolor{red}{3x}-\textcolor{green}{5})^2-25+4\\ \downarrow\\ \boxed{(3x-5)^2-21}$

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

Let's summarize the development steps, we'll do this now within the given equation:

$9x^2-30x+4=0 \\ (3x)^2-2\cdot 3x\cdot 5+4=0\\ (\textcolor{red}{3x})^2-2\cdot \textcolor{red}{3x}\cdot \textcolor{green}{5}\underline{\underline{+\textcolor{green}{5}^2-5^2}}+4=0\\ \downarrow\\ (\textcolor{red}{3x}-\textcolor{green}{5})^2-25+4=0\\ \downarrow\\ \boxed{(3x-5)^2-21=0}$

Let's note now that we are interested in the value of the expression:

$3x-5=\text{?}$

Therefore, we can return to the equation we got in the last stage and isolate this expression from it,

We'll do this by moving terms and taking the square root:

$(3x-5)^2-21=0\\ (3x-5)^2=21\hspace{6pt}\text{/}\sqrt{\hspace{6pt}}\\ \downarrow\\ \boxed{3x-5=\pm\sqrt{21}}$

(We'll remember of course that taking the square root from both sides of an equation involves considering two possibilities - with a positive and negative sign)

Therefore the correct answer is answer C.