Find the value of a and b:
√5-2/√5+2 - √5+2/√5-2 = a+b√5
Answers
Step-by-step explanation:
Hello friends!!
\frac{ \sqrt{5} - 2}{ \sqrt{5} + 2} - \frac{ \sqrt{5} + 2}{ \sqrt{5} - 2 } = a + b \sqrt{5}
5
+2
5
−2
−
5
−2
5
+2
=a+b
5
First we have to rationalise the denominator.
\frac{ \sqrt{5} - 2 }{ \sqrt{5} + 2 } \times \frac{ \sqrt{5} - 2}{ \sqrt{5} - 2 } - \frac{ \sqrt{5} + 2 }{ \sqrt{5} - 2} \times \frac{ \sqrt{5} + 2}{ \sqrt{5} + 2} = a + b \sqrt{5}
5
+2
5
−2
×
5
−2
5
−2
−
5
−2
5
+2
×
5
+2
5
+2
=a+b
5
\frac{ (\sqrt{5} - 2)( \sqrt{5} - 2) }{( \sqrt{5} + 2)( \sqrt{5} - 2) } - \frac{( \sqrt{5} + 2)( \sqrt{5} + 2) }{( \sqrt{5} -2)( \sqrt{5} + 2)} = a + b \sqrt{5}
(
5
+2)(
5
−2)
(
5
−2)(
5
−2)
−
(
5
−2)(
5
+2)
(
5
+2)(
5
+2)
=a+b
5
\frac{ {( \sqrt{5} - 2)}^{2} }{( \sqrt{5} - 2)( \sqrt{5} + 2) } - \frac{ {( \sqrt{5} + 2)}^{2} }{( \sqrt{5} + 2)( \sqrt{5} + 2) } = a + b \sqrt{5}
(
5
−2)(
5
+2)
(
5
−2)
2
−
(
5
+2)(
5
+2)
(
5
+2)
2
=a+b
5
Using identity:
( a - b )² = a² + b² - 2ab
( a + b )² = a² + b² + 2ab
( a - b )( a + b ) = a² - b²
\frac{ {( \sqrt{5} )}^{2} + {(2)}^{2} - 2 \times 2 \times \sqrt{5} }{ {( \sqrt{5} )}^{2} - {(2)}^{2} } - \frac{ {( \sqrt{5} )}^{2} + {(2)}^{2} + 2 \times 2 \times \sqrt{5} }{ {( \sqrt{5} )}^{2} - {(2)}^{2} } = a + b \sqrt{5}
(
5
)
2
−(2)
2
(
5
)
2
+(2)
2
−2×2×
5
−
(
5
)
2
−(2)
2
(
5
)
2
+(2)
2
+2×2×
5
=a+b
5
\frac{5 + 4 - 4 \sqrt{5} }{5 - 4} - \frac{5 + 4 + 4 \sqrt{5} }{5 - 4} = a + b \sqrt{5}
5−4
5+4−4
5
−
5−4
5+4+4
5
=a+b
5
\frac{9 - 4 \sqrt{5} }{1} - \frac{9 + 4 \sqrt{5} }{1} = a + b \sqrt{5}
1
9−4
5
−
1
9+4
5
=a+b
5
9 - 4 \sqrt{5} - (9 + 4\sqrt{5} ) = a + b \sqrt{5}9−4
5
−(9+4
5
)=a+b
5
9 - 4 \sqrt{5} - 9 - 4 \sqrt{5} = a + b \sqrt{5}9−4
5
−9−4
5
=a+b
5
- 8 \sqrt{5} = a + b \sqrt{5}−8
5
=a+b
5
Comparing these values,
a = 0
b = - 8
HOPE IT HELPS YOU...
Step-by-step explanation:
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