Cuthbert is conducting a chemistry experiment that calls for a number of chemicals to be mixed in various quantities. The one amount of which he is unsure is grams of potassium,p. IF cuthbert is certain that `(3p^2+14p+24)-2(p^2+7p+20)=0`. What is one possible value of 3p+6, the exact number of grams of potassium that Cuthbert would like to use for this experiment?

Options

A. 20
B. 18
C. 12
D. 10

Answer

To solve the problem step by step, we start with the equation given in the question:

1. **Write down the equation:**
\[ (3p^2 + 14p + 24) - 2(p^2 + 7p + 20) = 0 \]

2. **Distribute the -2 across the second term:**
\[ 3p^2 + 14p + 24 - 2p^2 - 14p - 40 = 0 \]

3. **Combine like terms:**
- Combine \(3p^2\) and \(-2p^2\):
\[ 3p^2 - 2p^2 = p^2 \]
- Combine \(14p\) and \(-14p\):
\[ 14p - 14p = 0 \]
- Combine \(24\) and \(-40\):
\[ 24 - 40 = -16 \]
- The equation simplifies to:
\[ p^2 - 16 = 0 \]

4. **Rearrange the equation:**
\[ p^2 = 16 \]

5. **Take the square root of both sides:**
\[ p = \pm 4 \]
This gives us two possible values for \(p\): \(4\) and \(-4\).

6. **Calculate \(3p + 6\) for both values of \(p\):**
- For \(p = 4\):
\[ 3(4) + 6 = 12 + 6 = 18 \]
- For \(p = -4\):
\[ 3(-4) + 6 = -12 + 6 = -6 \]

7. **Determine the valid amount of potassium:**
Since the amount of potassium cannot be negative, we discard \(p = -4\) and take \(p = 4\).

8. **Final answer:**
The possible value of \(3p + 6\) is:
\[ \boxed{18} \]