12(4x - 1/4x)square +5 (4x - 1/4x) -2

Dear Student,

Please find below the solution to the asked query:

We have : 12 ${\left(4 x- \frac{1}{4 x}\right)}^2$ + 5 $\left(4 x- \frac{1}{4 x}\right)$  -  2

Let , y  =  $\left(4 x- \frac{1}{4 x}\right)$  , So 

$\Rightarrow$12 y² + 5 y  -  2

$\Rightarrow$12 y² + 8 y  - 3 y -  2

$\Rightarrow$4 y ( 3 y + 2 ) - 1 ( 3 y + 2 )

$\Rightarrow$( 4 y - 1 ) ( 3 y + 2 ) , So 

Zeroes :  y  = $\frac{1}{4}$  and $-\frac{2}{3}$

But we assume y  =  $\left(4 x- \frac{1}{4 x}\right)$ , So

Case I : $\left(4 x- \frac{1}{4 x}\right)$ = $\frac{1}{4}$

$\Rightarrow \frac{16 x^{2 }- 1}{4 x} = \frac{1}{4}$
$\Rightarrow$16 x² -  1 =  x 

$\Rightarrow$16 x² - x -  1 = 0 

We know quadratic formula : x  = $\frac{- \mathrm{b} \pm \sqrt{{\mathrm{b}}^2 - 4 \mathrm{a} \mathrm{c}}}{2 \mathrm{a}}$
Here , a  =  16  , b  = - 1 and c = -  1 , So

x  = $\frac{-\left(- 1\right) \pm \sqrt{{\left(- 1\right)}^2 - 4 \times 16 \times \left(- 1\right)}}{2 \times 16} = \frac{1 \pm \sqrt{1+ 64}}{32} = \frac{1 \pm \sqrt{65}}{32}$ , So
Zeroes :  x = $\frac{\mathbf{1}\mathbf{ }\mathbf{+}\mathbf{ }\sqrt{\mathbf{65}}}{\mathbf{32}} \mathrm{and} \frac{\mathbf{1}\mathbf{ }\mathbf{-}\mathbf{ }\sqrt{\mathbf{65}}}{\mathbf{32}}$

Case II : $\left(4 x- \frac{1}{4 x}\right)$ = $-\frac{2}{3}$

$\Rightarrow \frac{16 x^{2 }- 1}{4 x} = - \frac{2}{3}$
$\Rightarrow$48 x² -  3 =  8 x 

$\Rightarrow$48 x² - 8 x -  3 = 0 

We know quadratic formula : x  = $\frac{- \mathrm{b} \pm \sqrt{{\mathrm{b}}^2 - 4 \mathrm{a} \mathrm{c}}}{2 \mathrm{a}}$
Here , a  =  48  , b  = - 8 and c = -  3 , So

x  = $\frac{-\left(- 8\right) \pm \sqrt{{\left(- 8\right)}^2 - 4 \times 48 \times \left(- 3\right)}}{2 \times 48} = \frac{8 \pm \sqrt{64+576}}{96} = \frac{8 \pm \sqrt{640}}{96} = \frac{8 \pm 8\sqrt{10}}{96} = \frac{1 \pm \sqrt{10}}{12}$ , So
Zeroes :  x = $\frac{\mathbf{1}\mathbf{ }\mathbf{+}\mathbf{ }\sqrt{\mathbf{10}}}{\mathbf{12}} \mathrm{and} \frac{\mathbf{1}\mathbf{ }\mathbf{-}\mathbf{ }\sqrt{\mathbf{10}}}{\mathbf{12}}$

Therefore,

Our solution : x = $\frac{\mathbf{1}\mathbf{ }\mathbf{+}\mathbf{ }\sqrt{\mathbf{65}}}{\mathbf{32}} \mathrm{and} \frac{\mathbf{1}\mathbf{ }\mathbf{-}\mathbf{ }\sqrt{\mathbf{65}}}{\mathbf{32}}$ and$\frac{\mathbf{1}\mathbf{ }\mathbf{+}\mathbf{ }\sqrt{\mathbf{10}}}{\mathbf{12}} \mathrm{and} \frac{\mathbf{1}\mathbf{ }\mathbf{-}\mathbf{ }\sqrt{\mathbf{10}}}{\mathbf{12}}$                                 ( Ans )


Hope this information will clear your doubts about topic.

If you have any more doubts just ask here on the forum and our experts will try to help you out as soon as possible.

Regards