Question

15.2 ^x+1 + 15.2 ^-x+2=135 then x=
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Answer 1

Given:

The expression:

$15.2 ^{x+1} + 15.2 ^{-x+2}=135$

To find:

The value of $x = ?$

Solution:

First of all, let us take 15 common from the given term:

$15(2^{x+1}+2^{-x+2}) =135\\\Rightarrow 2^{x+1}+2^{-x+2} = \dfrac{135}{15}\\\Rightarrow 2^{x+1}+2^{-x+2} = 9\\\Rightarrow 2.2^{x}+\dfrac{2^2}{2^x} = 9$

Now, let $y = 2^x$

Now, the given equation becomes:

$\Rightarrow 2y+\dfrac{2^2}{y} = 9\\\Rightarrow 2y^2+2^2 = 9y\\\Rightarrow 2y^2-9y+4 = 0$

Now, it has become a quadratic equation in $y$.

Let us solve this.

$\Rightarrow 2y^2-8y-y+4 = 0\\\Rightarrow 2y(y-4)-1(y-4) = 0\\\Rightarrow (2y-1)(y-4) = 0\\\Rightarrow y = \frac{1}{2}, 4$

Let us put the value of $y$ one by one:

$y = 2^x =\frac{1}{2}\\\Rightarrow 2^x=2^{-1}\\\Rightarrow x =-1$

$y = 2^x =4\\\Rightarrow 2^x=2^{2}\\\Rightarrow x =2$

So, the answer is:

$\bold{x=-1,2}$