Uncover valuable information and solutions with IDNLearn.com's extensive Q&A platform. Get prompt and accurate answers to your questions from our community of experts who are always ready to help.
Sagot :
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]
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]
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Your search for answers ends at IDNLearn.com. Thanks for visiting, and we look forward to helping you again soon.
