https://file.aiquickdraw.com/mathbot/file/1727179336984pr1eb1ep.jpg

The equation provided in the image is:

\[
\sqrt{12}x - \sqrt{5}x = \sqrt{6}
\]
The solution is given by:
\[
\frac{p\sqrt{2} + \sqrt{q}}{7}
\]
where \( p \) and \( q \) are integers, and you are tasked with finding the values of \( p \) and \( q \) without using a calculator.

### Step-by-step Solution:

1. **Simplify the equation:**
   The equation is:
   \[
   (\sqrt{12} - \sqrt{5})x = \sqrt{6}
   \]
   First, simplify the square roots:
   \[
   \sqrt{12} = 2\sqrt{3}, \quad \sqrt{5} = \sqrt{5}, \quad \text{so the equation becomes:}
   \]
   \[
   (2\sqrt{3} - \sqrt{5})x = \sqrt{6}
   \]

2. **Solve for \( x \):**
   To isolate \( x \), divide both sides of the equation by \( (2\sqrt{3} - \sqrt{5}) \):
   \[
   x = \frac{\sqrt{6}}{2\sqrt{3} - \sqrt{5}}
   \]

3. **Rationalize the denominator:**
   Multiply both the numerator and denominator by the conjugate of the denominator, \( (2\sqrt{3} + \sqrt{5}) \), to rationalize the denominator:
   \[
   x = \frac{\sqrt{6}(2\sqrt{3} + \sqrt{5})}{(2\sqrt{3} - \sqrt{5})(2\sqrt{3} + \sqrt{5})}
   \]
   The denominator simplifies as follows:
   \[
   (2\sqrt{3})^2 - (\sqrt{5})^2 = 12 - 5 = 7
   \]
   So the expression for \( x \) becomes:
   \[
   x = \frac{\sqrt{6}(2\sqrt{3} + \sqrt{5})}{7}
   \]

4. **Simplify the numerator:**
   Now simplify the numerator:
   \[
   \sqrt{6}(2\sqrt{3} + \sqrt{5}) = 2\sqrt{18} + \sqrt{30} = 6\sqrt{2} + \sqrt{30}
   \]
   Thus, the value of \( x \) is:
   \[
   x = \frac{6\sqrt{2} + \sqrt{30}}{7}
   \]

5. **Compare the result with the given form:**
   The solution is given in the form:
   \[
   \frac{p\sqrt{2} + \sqrt{q}}{7}
   \]
   By comparing, we can see that \( p = 6 \) and \( q = 30 \).

Thus, the values of \( p \) and \( q \) are:
\[
p = 6, \quad q = 30
\]

Would you like more details on any specific step?

### Related Questions:
1. How do you rationalize a denominator involving square roots?
2. Why is it necessary to use the conjugate when rationalizing denominators?
3. Can you solve a similar equation where different square roots are involved?
4. How do you simplify complex square root expressions?
5. What are the properties of square roots and radicals that help in solving equations?

### Tip:
Always check your final expression carefully to ensure it matches the form of the solution provided in the question.

The solution of the equation \( \sqrt{12} x - \sqrt{5} x = \sqrt{6} \) is \( \frac{p \sqrt{2} + \sqrt{q}}{7} \), where p and q are constants. Without using a calculator, find the values of integer p and q.

Discover how to solve the radical equation \( \sqrt{12}x - \sqrt{5}x = \sqrt{6} \) and determine the integer values of p and q, using the method of rationalizing the denominator and simplifying square roots.

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