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Question Number 187357 by cortano12 last updated on 16/Feb/23

Answered by Humble last updated on 16/Feb/23

3(√5)(√(2x+3)) +2(√5)(√(3x+2)) = (√(3x+2))(√(2x+3))    21600x^2 +46800x+21600=36x^4 −1644x^3 +16741x^2 +46306x+28561    36x^4 −1644x^3 −4859x^2 −494x+6961=0    (x−1)(36x^3 −1608x^2 −6467x−6961)=0  x=1or 36x^3 −1608x^2 −6467x−6961=0  but x≠1  36x^3 −1608x^2 −6467x−6961=0  Applying N−R Method  x≈48.45625...

$$\mathrm{3}\sqrt{\mathrm{5}}\sqrt{\mathrm{2}{x}+\mathrm{3}}\:+\mathrm{2}\sqrt{\mathrm{5}}\sqrt{\mathrm{3}{x}+\mathrm{2}}\:=\:\sqrt{\mathrm{3}{x}+\mathrm{2}}\sqrt{\mathrm{2}{x}+\mathrm{3}} \\ $$$$ \\ $$$$\mathrm{21600}{x}^{\mathrm{2}} +\mathrm{46800}{x}+\mathrm{21600}=\mathrm{36}{x}^{\mathrm{4}} −\mathrm{1644}{x}^{\mathrm{3}} +\mathrm{16741}{x}^{\mathrm{2}} +\mathrm{46306}{x}+\mathrm{28561} \\ $$$$ \\ $$$$\mathrm{36}{x}^{\mathrm{4}} −\mathrm{1644}{x}^{\mathrm{3}} −\mathrm{4859}{x}^{\mathrm{2}} −\mathrm{494}{x}+\mathrm{6961}=\mathrm{0} \\ $$$$ \\ $$$$\left({x}−\mathrm{1}\right)\left(\mathrm{36}{x}^{\mathrm{3}} −\mathrm{1608}{x}^{\mathrm{2}} −\mathrm{6467}{x}−\mathrm{6961}\right)=\mathrm{0} \\ $$$${x}=\mathrm{1}{or}\:\mathrm{36}{x}^{\mathrm{3}} −\mathrm{1608}{x}^{\mathrm{2}} −\mathrm{6467}{x}−\mathrm{6961}=\mathrm{0} \\ $$$${but}\:{x}\neq\mathrm{1} \\ $$$$\mathrm{36}{x}^{\mathrm{3}} −\mathrm{1608}{x}^{\mathrm{2}} −\mathrm{6467}{x}−\mathrm{6961}=\mathrm{0} \\ $$$${Applying}\:{N}−{R}\:{Method} \\ $$$${x}\approx\mathrm{48}.\mathrm{45625}... \\ $$

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