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.
