Question

Give me solution of this question plz

Answer 1

Hey there!

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

Let 4^x = m.

4(m + 1/m) = 10

4(m² + 1) = 10m

4m² - 10m + 4 = 0

2m² - 5m + 2 = 0

2m² -4m - m + 2 = 0

2m ( m - 2) - 1 ( m - 2 ) = 0

( 2m - 1 ) ( m - 2 ) = 0

2m = 1 , m = 2

m = 1/2 , m = 2

Taking m = 1/2

4^x = 1/2

2^2x = 2^-1

2x = -1

x = -1/2

or,

m = 2

4^x = 2

2^2x = 2

2x = 1

x = 1/2

Therefore , x = 1/2 or -1/2

Answer 2

Here is the answer

The answer to the question is given in the attachment.

Next steps are down here

$\bf{2k - 1 = 0 \: \: \: \: \: or \: \: \: k - 2 = 0}$

$\bf{2k = 1 \: \: \: \: \: \: \: \: or \: \: \: \: \: \: \: \: \: \: \: k = 2}$


$\bf{k = \frac{1}{2} \: \: \: \: \: \: \: \: \: or \: \: \: \: \: \: \: k = 2}$


Now,
$\bf{4 {}^{x} = \frac{1}{2} } \\ \\ \bf{ 2 {}^{2x} = {2}^{ - 1} }$

$\bf{2x = - 1}$

$\bf{x = \frac{ - 1}{2}}$

$\bf{Or }$

$\bf{4 {}^{x} = 2} \\ \\ \bf{(2 {}^{2} ) {}^{x} = 2}$


$\bf{2x = 1} \\ \\ \bf{ x = \frac{1}{2} }$

Thus ,

We got 2 values of x

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

Thanks!!