Question

if, (5 + 2√6)^x + (5-2√6)^x = 10
find the value
of x.​

Answer 1

Answer:

x=1 or -1

Step-by-step explanation:

put x=1 in the equation

(5 + 2√6)^x + (5-2√6)^x = 10

(5 + 2√6)+ (5-2√6) = 10

10=10

and we can clearly see that

$(5 + 2 \sqrt{6} ) \times (5 - 2 \sqrt{6} )$

$= {5}^{2} - {(2 \sqrt{6}) }^{2}$

$= 25 - 24 \\ = 1$

this means 5+2sqrt(6) and 5-2sqrt(6)

are multiplicative inverse of each other

this means

$(5 + 2 \sqrt{6} ) ^{ - 1} = (5 - 2 \sqrt{6} )$

so putting x=-1 in equation

(5 + 2√6)^x + (5-2√6)^x = 10

(5 + 2√6)^-1 + (5-2√6)^-1 = 10

(5 -2√6)+ (5+2√6) = 10

10=10

hence x= 1 or -1