Question

if 5y + 1/5y = 6, find 5y - 1/5y.

Answer 1

assume 5y =x
then,
x+1/x=6
square both sides
find x2+1/x2=34
then x-1/x whole square= x2+1/x2-2= 32
so x-1/x= root 32

Answer 2

solution:-

=> 5y + 1/5y = 6

Squaring both sides.

=> [ 5y + 1/5y ]² = [ 6 ]²

[ 5y + 1/5y ]² compare with [ a + b ]²

[a + b ]² = a² + 2ab + b²

so,

=> (5y)² + 2(5y × 1/5y) + (1/5y)² = 36

=> 25y² + 2 + 1/25y² = 36

=> 25y² + 1/5y² + 2 = 36

=> 25y² + 1/5y² = 36 - 2

=> 25y² + 1/5y² = 34

substracing 2 both sides.

=> 25y² + 1/5y² - 2 = 34 - 2

=> 25y² - 2 + 1/5y² = 32

25y² - 2 + 1/5y² compare with a²-2ab + b²

so, we know a²-2ab + b² = [a - b]²

hence,

=> [ 5y - 1/5y]² = 32

=> 5y - 1/5y = √32

=> 5y - 1/5y = √2×2×2×2×2

=> 5y - 1/5y = 4√2 or 2√8

i hope it helps you.