Question

$\large{ \green{ \boxed{ \bold{ \red{ \: QUESTION :}}}}}$
If x = (1 + √2 + √3) Then Find the value of 2x⁴ - 8x³ - 5x² +26x - 28.

Given Options Are :
1 ) √6
2 ) 2√3
3 ) 3√2
4 ) 6√6

$\large{ \blue{ \boxed{ \bold{{ \pink{ \: Thank \: You \: !}}}}}}$
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Answer 1

AnswEr :

$\underline{\bigstar\:\textsf{According to the Question Now :}}$

$:\implies\tt x = 1+\sqrt{2}+\sqrt{3}\\\\\\:\implies\tt x-1=\sqrt{2}+\sqrt{3}\\\\{\scriptsize\qquad\bf{\dag}\:\textsf{Squaring Both Sides}}\\\\:\implies\tt (x-1)^2=(\sqrt{2}+\sqrt{3})^2\\\\\\:\implies\tt x^2+1^2-(2.x.1)=(\sqrt{2})^2+(\sqrt{3})^2+(2.\sqrt{2}.\sqrt{3})\\\\\\:\implies\tt x^2+1-2x=2+3+2\sqrt{6}\\\\\\:\implies\tt x^2-2x-4=2\sqrt{6}\qquad\dfrac{\qquad}{}eq.(1)\\\\{\scriptsize\qquad\bf{\dag}\:\textsf{Squaring Both Sides}}\\\\:\implies\tt (x^2-2x-4)^2=(2\sqrt{6})^2$

$\\:\implies\tt (x^2)^2+(-2x)^2+(-4)^2+2[(x^2.-2x)+(-2x.-4)+(-4.x^2)]=(2.2.6)\\\\\\:\implies\tt x^4+4x^2+16+2[-2x^3+8x-4x^2]=24\\\\\\:\implies\tt x^4+4x^2+16-4x^3+16x-8x^2-24=0\\\\\\:\implies\tt x^4-4x^3-4x^2+16x-8=0\\\\{\scriptsize\qquad\bf{\dag}\:\textsf{Multiplying by 2 Both Sides}}\\\\:\implies\tt 2x^4-8x^3-8x^2+32x-16=0\\\\\\:\implies\tt2x^4-8x^3-5x^2-3x^2+26x+6x-16-12+12=0\\\\\\:\implies\tt(2x^4-8x^3-5x^2+26x-28)-3x^2+6x+12=0\\\\\\:\implies\tt2x^4-8x^3-5x^2+26x-28=3x^2-6x+12$

$\\:\implies\tt2x^4-8x^3-5x^2+26x-28=3(x^2-2x+4)\\\\{\scriptsize\qquad\bf{\dag}\:\textsf{Putting the value from eq.(1)}}\\\\:\implies\tt2x^4-8x^3-5x^2+26x-28=3(2\sqrt{6})\\\\\\:\implies\boxed{\tt2x^4-8x^3-5x^2+26x-28=6\sqrt{6}}$

⠀

$\therefore\:\underline{\textsf{Required Value of the equation is \textbf{6 \sqrt{ \textbf6} }}}$

Answer 2

$\huge{\underline{\red{\mathcal{AnsWer}}}}$

$= > 6 \sqrt{6}$

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Refer to attachment for solution

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