To solve the problem of finding the values of [tex]\( p \)[/tex] and [tex]\( q \)[/tex] for the equation [tex]\(\frac{3+2 \sqrt{5}}{3-2 \sqrt{5}} = p + q \sqrt{5}\)[/tex], follow these steps:
1. Rationalize the Denominator:
To remove the square root from the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\(3 - 2 \sqrt{5}\)[/tex] is [tex]\(3 + 2 \sqrt{5}\)[/tex].
[tex]\[
\frac{3 + 2 \sqrt{5}}{3 - 2 \sqrt{5}} \times \frac{3 + 2 \sqrt{5}}{3 + 2 \sqrt{5}}
\][/tex]
2. Expand the Numerator:
The numerator becomes:
[tex]\[
(3 + 2 \sqrt{5})(3 + 2 \sqrt{5}) = 3^2 + 2 \cdot 3 \cdot 2 \sqrt{5} + (2 \sqrt{5})^2
\][/tex]
Calculate each term:
[tex]\[
3^2 = 9
\][/tex]
[tex]\[
2 \cdot 3 \cdot 2 \sqrt{5} = 12 \sqrt{5}
\][/tex]
[tex]\[
(2 \sqrt{5})^2 = 4 \cdot 5 = 20
\][/tex]
Adding these together:
[tex]\[
9 + 12 \sqrt{5} + 20 = 29 + 12 \sqrt{5}
\][/tex]
3. Expand the Denominator:
The denominator becomes:
[tex]\[
(3 - 2 \sqrt{5})(3 + 2 \sqrt{5}) = 3^2 - (2 \sqrt{5})^2
\][/tex]
Calculate each term:
[tex]\[
3^2 = 9
\][/tex]
[tex]\[
(2 \sqrt{5})^2 = 4 \cdot 5 = 20
\][/tex]
Subtract these:
[tex]\[
9 - 20 = -11
\][/tex]
4. Form the Fraction:
Hence, we have:
[tex]\[
\frac{29 + 12 \sqrt{5}}{-11}
\][/tex]
5. Separate into Two Fractions:
Write this expression as separate fractions:
[tex]\[
\frac{29}{-11} + \frac{12 \sqrt{5}}{-11}
\][/tex]
Simplify each term:
[tex]\[
\frac{29}{-11} = -\frac{29}{11}
\][/tex]
[tex]\[
\frac{12 \sqrt{5}}{-11} = -\frac{12}{11} \sqrt{5}
\][/tex]
6. Determine [tex]\( p \)[/tex] and [tex]\( q \)[/tex]:
Comparing this to the form [tex]\( p + q \sqrt{5} \)[/tex], we identify:
[tex]\[
p = -\frac{29}{11}
\][/tex]
[tex]\[
q = -\frac{12}{11}
\][/tex]
Thus, the values of [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are:
[tex]\[
p = -\frac{29}{11}, \quad q = -\frac{12}{11}
\][/tex]
