please help solve x(5^1/2-2)^x+(5^1/2+2)^x=18

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should notice that if you multiply the bases of exponentials yields a difference of squares such that:

`(sqrt5-2)(sqrt5+2) = 5- 4 = 1`

Hence, `sqrt5+2 = 1/(sqrt5-2)`

You should come up with the substitution (sqrt5+2)^x = y such that:

`y + 1/y = 18`

You need to move the terms to the left side and you need to bring therms to a common denominator such that:

`y^2 - 18y + 1 = 0`

You should use quadratic formula to find `y_1`  and `y_2`  such that:

`y_(1,2) = (18+-sqrt(324 - 4))/2`

`y_(1,2) = (18+-sqrt(320))/2 =gt y_(1,2) = (18+-8sqrt(5))/2`

`y_(1,2) = 9 +- 4sqrt5`

You need to find x, hence you need to solve the equations `(sqrt5+2)^x = y_(1,2)`

Substituting  `9 +- 4sqrt5`  for `y_(1,2)`  yields:

`(sqrt5+2)^x = 9 + 4sqrt5 =gt ln (sqrt5+2)^x = ln(9 + 4sqrt5)`

`x_1 = (ln(9 + 4sqrt5))/(ln (sqrt5+2))`

`x_2 =(ln(9- 4sqrt5))/(ln (sqrt5+2))`

Hence, the solutions to exponential equation are `x_1 = (ln(9 + 4sqrt5))/(ln (sqrt5+2)) ` and `x_2 = (ln(9 - 4sqrt5))/(ln (sqrt5+2)).`

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