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Question
If `[ 2 + sqrt5 ]/[ 2 - sqrt5] = x and [2 - sqrt5 ]/[ 2 + sqrt5] = y`; find the value of x2 - y2.
Solution
x =`[ 2 + sqrt5 ]/[ 2 - sqrt5]`
=`[ 2 + sqrt5 ]/[ 2 - sqrt5] xx [ 2 + sqrt5 ]/[ 2 + sqrt5] `
=`[( 2 + sqrt5)^2]/[2^2 - (sqrt5)^2]`
=`( 4 + 4sqrt5 + 5)/(4 - 5)`
=`( 9 + 4sqrt5)/ -1`
= `- ( 9 - 4sqrt5)`
y = `[2 - sqrt5 ]/[ 2 + sqrt5]`
=`[ 2 - sqrt5 ]/[ 2 + sqrt5] xx [ 2 - sqrt5 ]/[ 2 - sqrt5] `
=`[( 2 - sqrt5)^2]/[2^2 - (sqrt5)^2]`
=`( 4 - 4sqrt5 + 5)/(4 - 5)`
=`( 9 - 4sqrt5 )/ -1`
=` - ( 9 + 4sqrt5 )`
∴ ` x^2 - y^2 = ( - 9 - 4sqrt5 )^2 - ( - 9 + 4sqrt5 )^2`
= `81 + 72sqrt5 + 80 - ( 81 - 72sqrt5 + 80)`
= `81 + 72sqrt5 + 80 - 81 + 72sqrt5 - 80`
= `144sqrt5`
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