Completing the Square with 16x²+24x-40=0: Finding the Value of 12x+9

First, let's recall the principles of the "completing the square" method and its general concept:

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 perfect square formulas:

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

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

In the given problem we will first look at the given equation:

$16x^2+24x-40=0$

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 addition form of the perfect square formula, since the non-square term in the given expression, $24x$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 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:

$\underline{16x^2+24x}-40 \textcolor{blue}{\leftrightarrow} \underline{ c^2+2cd+d^2 }\\ \downarrow\\ \underline{(\textcolor{red}{4x})^2\stackrel{\downarrow}{+2 }\cdot \textcolor{red}{4x}\cdot \textcolor{green}{3}}-40 \textcolor{blue}{\leftrightarrow} \underline{ \textcolor{red}{c}^2\stackrel{\downarrow}{+2 }\textcolor{red}{c}\textcolor{green}{d}\hspace{2pt}\boxed{+\textcolor{green}{d}^2}} \\$We can notice that compared to the perfect square formula (on the right side of the blue press in the previous calculation) we are actually making the analogy:

$\begin{cases} 4x\textcolor{blue}{\leftrightarrow}c\\ 3\textcolor{blue}{\leftrightarrow}d \end{cases}$

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

We will need to add to these two terms the term$3^2$, but we don't want to change the value of the expression in question, so we'll also subtract this term from the expression,

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

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

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

$(4x)^2+2\cdot 4x\cdot 3-40\\ (4x)^2+2\cdot 4x\cdot 3\underline{\underline{+3^2-3^2}}-40\\ (\textcolor{red}{4x})^2+2\cdot \textcolor{red}{4x}\cdot \textcolor{green}{3}+\textcolor{green}{3}^2-9-40\\ \downarrow\\ (\textcolor{red}{4x}+\textcolor{green}{3})^2-9-40\\ \downarrow\\ \boxed{(4x+3)^2-49}$

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:

$16x^2+24x-40=0\\ (4x)^2+2\cdot 4x\cdot 3\underline{\underline{+3^2-3^2}}-40=0\\ (\textcolor{red}{4x})^2+2\cdot \textcolor{red}{4x}\cdot \textcolor{green}{3}+\textcolor{green}{3}^2-9-40=0\\ \downarrow\\ (\textcolor{red}{4x}+\textcolor{green}{3})^2-9-40=0\\ \downarrow\\ \boxed{(4x+3)^2-49=0}$

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

$12x+9=\text{?}$

But how is this expression related to the expression we got in the last stage?

We can notice the connection if we also refer to the fact that the expression whose value we want to calculate can be factored by taking out a common factor to get the expression:

$$$12x+9=\text{?} \\ \downarrow\\ 3(4x+3)=\text{?}$

That is, to answer the question asked, it's enough for us to find the value of the expression:

$4x+3=\text{?}$

But the value of this expression we can easily get from the equation we got in the last stage (that is, after performing the "completing the square"), we'll do this by isolating the squared expression on one side and then taking the square root:

$(4x+3)^2-49=0\\ (4x+3)^2=49\hspace{6pt}\text{/}\sqrt{\hspace{6pt}}\\ \downarrow\\ 4x+3=\pm7$

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

Therefore, now we can complete calculating the value of the requested expression, we'll do this by substituting the value of the expression we just calculated (highlighted in color in the next calculation) in the expression whose value we want to calculate:

$12x+9= \\ 3(\textcolor{orange}{4x+3})=\\ 3\cdot(\textcolor{orange}{\pm7})=\\ \pm21 \\ \downarrow\\ \boxed{12x+9\stackrel{\textcolor{red}{!}}{=}\pm21}$

Therefore the correct answer is answer A.