Alright, let's solve the equation step-by-step to find the correct solutions.
The given equation is:
[tex]\[
\frac{A}{6} = (s + 2)^2
\][/tex]
To isolate [tex]\( s \)[/tex], we need to take the square root of both sides of the equation. When we take the square root of both sides, we need to remember that we get both the positive and negative roots. Thus, we have:
[tex]\[
\sqrt{\frac{A}{6}} = s + 2 \quad \text{or} \quad -\sqrt{\frac{A}{6}} = s + 2
\][/tex]
Now, we isolate [tex]\( s \)[/tex]:
For the positive root,
[tex]\[
s = \sqrt{\frac{A}{6}} - 2
\][/tex]
For the negative root,
[tex]\[
s = -\sqrt{\frac{A}{6}} - 2
\][/tex]
So our solutions for [tex]\( s \)[/tex] are:
[tex]\[
s = \sqrt{\frac{A}{6}} - 2 \quad \text{and} \quad s = -\sqrt{\frac{A}{6}} - 2
\][/tex]
In the format given in your prompt, these solutions can be written as:
[tex]\[
\sqrt{\frac{A}{6}} - 2 = s \quad \text{and} \quad -\sqrt{\frac{A}{6}} - 2 = s
\][/tex]
Thus, plugging in the values into the format [tex]\(\square \pm \frac{\square}{\square}=s\)[/tex]:
[tex]\[
s = \sqrt{\frac{A}{6}} - 2 \quad \text{or} \quad s = -2 \pm \sqrt{\frac{A}{6}}
\][/tex]
Therefore,
[tex]\[
\sqrt{\frac{A}{6}} - 2 = s
\][/tex]
and
[tex]\[
-2 \pm \frac{\sqrt{6}\sqrt{A}}{6}=s
\][/tex]
Now let's write each value in its respective boxes:
- The first box for the constant term [tex]\(2\)[/tex]: [tex]\(2\)[/tex]
- The second box value is the numerator of the fraction: [tex]\(\sqrt{6}\sqrt{A}\)[/tex]
- The third box value is the denominator of the fraction: [tex]\(6\)[/tex]
Thus:
[tex]\(\square \pm \frac{\square}{\square} = s = 2 \pm \frac{\sqrt{6}\sqrt{A}}{6} \)[/tex]
