Solving 25x²+30x+6=0: Complete the Square Method

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

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

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 a form of a quadratic binomial,

That is, we use the formulas for quadratic binomial:

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

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

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

$25x^2+30x+6=0$

First, we'll try to give the expression on the left side of the equation a form that resembles the right side of the quadratic binomial formulas, we also identify that we are interested in the addition form of the quadratic binomial formula, since the non-squared term in the given expression, 30x, has a positive 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 quadratic binomial formula,

For this - first we'll present these terms in a form similar to the form of the first two terms in the quadratic binomial formula:

$\underline{ 25x^2+30x}+6\textcolor{blue}{\leftrightarrow} \underline{ c^2+2cd+d^2 }\\ \downarrow\\ \underline{(\textcolor{red}{5x})^2\stackrel{\downarrow}{+2 }\cdot \textcolor{red}{5x}\cdot \textcolor{green}{3}}+6 \textcolor{blue}{\leftrightarrow} \underline{ \textcolor{red}{c}^2\stackrel{\downarrow}{+2 }\textcolor{red}{c}\textcolor{green}{d}\hspace{2pt}\boxed{+\textcolor{green}{d}^2}} \\$

We can notice that in comparison to the quadratic binomial formula (which is on the right side of the blue arrow in the previous calculation) we are actually making the analogy:

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

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

We will need to add to these two terms the term

$3^2$

However, 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 quadratic binomial form,

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

Next - we'll put into quadratic binomial form the appropriate expression (demonstrated with colors) and in the final stage we'll further simplify the expression:

$(5x)^2+2\cdot 5x\cdot 3+6\\ (5x)^2+2\cdot5x\cdot 3\underline{\underline{+3^2-3^2}}+6\\ (\textcolor{red}{5x})^2+2\cdot \textcolor{red}{5x}\cdot \textcolor{green}{3}+\textcolor{green}{3}^2-9+6\\ \downarrow\\ (\textcolor{red}{5x}+\textcolor{green}{3})^2-9+6\\ \downarrow\\ \boxed{(5x+3)^2-3}$

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

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

$25x^2+30x+6=0 \\ (5x)^2+2\cdot 5x\cdot 3+6=0\\ (\textcolor{red}{5x})^2+2\cdot \textcolor{red}{5x}\cdot \textcolor{green}{3}\underline{\underline{+\textcolor{green}{3}^2-3^2}}+6=0\\ \downarrow\\ (\textcolor{red}{5x}+\textcolor{green}{3})^2-9+6=0\\ \downarrow\\ \boxed{(5x+3)^2-3=0}$

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

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

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:

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

(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 D.