Question

5^(x+1) + 5^(2-x) = 5^3 + 1
find x​

Answer 1

$\sf 5^{(x+1)} + 5^{(2-x)} = 5^3 + 1$

$\sf {5}^{x} \times 5 + {5}^{2} \times {5}^{ - x} = 126$

$\sf {5}^{x} \times 5 + \dfrac{25}{ {5}^{x} } = 126$

$\sf substitute \: {5}^{x} = y$

$\sf y \times 5 + \dfrac{25}{y} = 126$

$\sf 5y + \dfrac{25}{y} = 126$

$\sf \dfrac{ {5y}^{2} + 25}{y} = 126$

$\sf {5y}^{2} + 25 = 126y$

$\sf 5y^{2} - 126y + 25 = 0$

$\sf 5y^{2} - 125y - y+ 25 = 0$

$\sf 5y(y - 25) - (y - 25) = 0$

$\sf (y - 25)(5y - 1) = 0$

$\sf y = 25 \: or \: \cfrac{1}{5}$

$\sf {5}^{x} = y$

If y = 25

$\sf {5}^{x} = 25$

$\sf {5}^{x} = {5}^{2}$

$\boxed{ \sf x = 2}$

If y = 1/5

$\sf {5}^{x} = \dfrac{1}{5}$

$\sf {5}^{x} = {5}^{ - 1}$

$\boxed{ \sf x = - 1}$

Hence, x = 2 or -1