Question

4 power of x plus one plus 4power of one minus x equals 10 then find the vale of x​

Answer 1

Given,

${4}^{(x + 1)} + {4}^{(1 - x)} = 10$

$( {4}^{x} \times {4}^{1}) + ( {4}^{1} \times {4}^{ - x} ) = 10$

$4( {4}^{x} + {4}^{ - x} ) = 10$

$4( {4}^{x} + \frac{1}{ {4}^{x} } ) = 10$

$2( \frac{ {4}^{2x} + 1 }{ {4}^{x} } ) = 5$

Let us assume,

${4}^{x} = z$

Then,

$2( \frac{ {z}^{2} + 1 }{z} ) = 5$

$\frac{2 {z}^{2} + 2}{z} = 5$

$2 {z}^{2} + 2 = 5z$

$2 {z}^{2} - 5z + 2 = 0$

$2 {z}^{2} - 4z - z + 2 = 0$

$2z(z - 2) - 1(z - 2) = 0$

$(2z - 1)(z - 2) = 0$

$z = \frac{1}{2} \: \: (or )\: \: z = 2$

i.e,

${4}^{x} = 2 \: \: (or )\: \: {4}^{x} = \frac{1}{2}$

${2}^{2x} = 2 \: \: (or) \: \: {2}^{2x} = {2}^{ - 1}$

$2x = 1 \: \: (or )\: \: 2x = - 1$

$x = \frac{1}{2} \: \: (or )\: \: x = \frac{ - 1}{2}$