Question

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Answer 1

$\texttt{\textsf{\large{\underline{Question 1}:}}}$

Given:

$\sf \implies {3}^{ {x}^{2} }:{3}^{2x} = 27:1$

We have to find the values of x.

$\sf \implies \dfrac{ {3}^{ {x}^{2} } }{ {3}^{2x} } = \dfrac{27}{1}$

As we know that:

$\sf \implies \dfrac{ {x}^{a} }{ {x}^{b} } = {x}^{a - b}$

Therefore:

$\sf \implies {3}^{ {x}^{2} - 2x} = 27$

$\sf \implies {3}^{ {x}^{2} - 2x} = {3}^{3}$

Comparing base, we get:

$\sf \implies {x}^{2} - 2x = 3$

$\sf \implies {x}^{2} - 2x - 3 = 0$

By splitting the middle term, we get:

$\sf \implies {x}^{2} - 3x + x- 3 = 0$

$\sf \implies x(x - 3) + 1(x- 3 )= 0$

$\sf \implies (x+ 1)(x- 3 )= 0$

Therefore:

$\implies \begin{cases} \sf x + 1 = 0 \\ \sf x - 3 = 0 \end{cases}$

$\sf \implies x = -1,3 \: \: \: \: \: (Answer)$

$\texttt{\textsf{\large{\underline{Question 2}:}}}$

Given:

$\sf \implies {144}^{ {x}^{2} - 2 } - {12}^{3x - 2} = 0$

We have to find out the values of x.

The given equation can be written as:

$\sf \implies {144}^{ {x}^{2} - 2 } = {12}^{3x - 2}$

$\sf \implies {( {12}^{2} )}^{ {x}^{2} - 2 } = {12}^{3x - 2}$

$\sf \implies {(12)}^{ 2{x}^{2} -4} = {12}^{3x - 2}$

Comparing base, we get:

$\sf \implies2 {x}^{2} - 4 = 3x - 2$

$\sf \implies2 {x}^{2} - 3x - 2 = 0$

By splitting the middle term, we get:

$\sf \implies2 {x}^{2} - 4x + x - 2 = 0$

$\sf \implies 2x(x -2) + 1(x - 2) = 0$

$\sf \implies (2x+ 1)(x - 2) = 0$

Therefore:

$\implies \begin{cases} \sf2 x + 1 = 0 \\ \sf x - 2 = 0 \end{cases}$

$\sf \implies x = \dfrac{ - 1}{2} ,2 \: \: \: \: \: (Answer)$