Answer :

To find the values of [tex]\( p \)[/tex] and [tex]\( q \)[/tex] such that

[tex]\[ \frac{3 + 2 \sqrt{5}}{3 - 2 \sqrt{5}} = p + q \sqrt{5}, \][/tex]

follow these steps:

1. Identify the expression:
Let the given fraction be [tex]\( A \)[/tex]:

[tex]\[ A = \frac{3 + 2 \sqrt{5}}{3 - 2 \sqrt{5}}. \][/tex]

2. Rationalize the denominator:
Multiply both the numerator and the denominator by the conjugate of the denominator, which is [tex]\( 3 + 2 \sqrt{5} \)[/tex]:

[tex]\[ A = \frac{(3 + 2 \sqrt{5})(3 + 2 \sqrt{5})}{(3 - 2 \sqrt{5})(3 + 2 \sqrt{5})}. \][/tex]

3. Simplify the denominator:
Use the difference of squares formula [tex]\( (a - b)(a + b) = a^2 - b^2 \)[/tex]:

[tex]\[ (3 - 2 \sqrt{5})(3 + 2 \sqrt{5}) = 3^2 - (2 \sqrt{5})^2 = 9 - 4 \times 5 = 9 - 20 = -11. \][/tex]

4. Simplify the numerator:
Expand [tex]\( (3 + 2 \sqrt{5})^2 \)[/tex]:

[tex]\[ (3 + 2 \sqrt{5})^2 = 3^2 + 2 \cdot 3 \cdot 2 \sqrt{5} + (2 \sqrt{5})^2 = 9 + 12 \sqrt{5} + 4 \times 5 = 9 + 12 \sqrt{5} + 20 = 29 + 12 \sqrt{5}. \][/tex]

5. Combine both parts:
Substitute the simplified numerator and denominator back into the fraction:

[tex]\[ A = \frac{29 + 12 \sqrt{5}}{-11}. \][/tex]

6. Separate into real and irrational parts:
Rewrite the fraction by separating the real and irrational parts:

[tex]\[ A = \frac{29}{-11} + \frac{12 \sqrt{5}}{-11} = -\frac{29}{11} - \frac{12}{11} \sqrt{5}. \][/tex]

7. Compare with the form [tex]\( p + q \sqrt{5} \)[/tex]:
By comparison, we get:

[tex]\[ p = -\frac{29}{11}, \quad q = -\frac{12}{11}. \][/tex]

8. Final values for [tex]\( p \)[/tex] and [tex]\( q \)[/tex]:
Therefore, the values of [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are:

[tex]\[ p = -2.6363636363636362, \quad q = -1.0909090909090908. \][/tex]

Thus, the values of [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are [tex]\( -2.6363636363636362 \)[/tex] and [tex]\( -1.0909090909090908 \)[/tex], respectively.