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Question Number 76122 by benjo last updated on 24/Dec/19

	Answered by john santuy last updated on 24/Dec/19

	$i t h i n k t h i s p r o b l e m n o t d i f f i c u l t$
	Commented by $@ty@m123 last updated on 24/Dec/19

	$T h e t e r m ‘ ‘ {d i f f i c u l t}^{″} i s r a t i o n a l .$ $I t v a r i e s f r o m p e r s o n t o p e r s o n .$
	Commented by john santuy last updated on 24/Dec/19

	$y e s s i r . t h a n k s y o u$
	Answered by MJS last updated on 24/Dec/19

	$x + 4 - \frac{3}{x - 4} = 2 \sqrt{\frac{x + 4}{x - 4}}$ $\frac{x^{2} - 19}{x - 4} = 2 \sqrt{\frac{x + 4}{x - 4}}$ $squaring \Rightarrow not all solutions will fit the given$ $equation$ ${(\frac{x^{2} - 19}{x - 4})}^{2} = 4 \frac{x + 4}{x - 4}$ ${(x^{2} - 19)}^{2} = 4 (x^{2} - 16)$ $x^{4} - 42 x^{2} + 425 = 0$ $x^{2} = 25 \lor x^{2} = 17$ $\Rightarrow x = \pm 5 \lor x = \pm \sqrt{17}$ $testing in given equation$ $\Rightarrow x = 5 \lor x = - \sqrt{17}$
	Commented by benjo last updated on 24/Dec/19

	$thanks you sir$
	Answered by behi83417@gmail.com last updated on 24/Dec/19
	$((x+4)/(x−4))=t^2 ⇒((2x)/8)=((t^2 +1)/(t^2 −1))⇒x=4((t^2 +1)/(t^2 −1)) 4((t^2 +1)/(t^2 −1))+4−2t=(3/(4((t^2 +1)/(t^2 −1))−4))=((3(t^2 −1))/(4(2))) ((8t^2 )/(t^2 −1))−2t=((3(t^2 −1))/8) 64t^2 −16t(t^2 −1)=3(t^2 −1)^2 ⇒3t^4 +16t^3 −70t^2 −16t+3=0 ⇒t=−8.123,−(1/3),+0.123,+3.0 { ((((x+4)/(x−4))=66⇒(x/4)=((67)/(65))⇒x=4.123)),((((x+4)/(x−4))=(1/9)⇒(x/4)=−((10)/8)⇒x=−5.0)),((((x+4)/(x−4))=0.015⇒(x/4)=((1.015)/(0.985))⇒x=−4.123)),((((x+4)/(x−4))=9⇒(x/4)=((10)/8)⇒x=+5.0)) :}$
	$\frac{x + 4}{x - 4} = t^{2} \Rightarrow \frac{2 x}{8} = \frac{t^{2} + 1}{t^{2} - 1} \Rightarrow x = 4 \frac{t^{2} + 1}{t^{2} - 1}$ $4 \frac{t^{2} + 1}{t^{2} - 1} + 4 - 2 t = \frac{3}{4 \frac{t^{2} + 1}{t^{2} - 1} - 4} = \frac{3 (t^{2} - 1)}{4 (2)}$ $\frac{{8 t}^{2}}{t^{2} - 1} - 2 t = \frac{3 (t^{2} - 1)}{8}$ ${64 t}^{2} - 16 t (t^{2} - 1) = 3 {(t^{2} - 1)}^{2}$ $\Rightarrow {3 t}^{4} + {16 t}^{3} - {70 t}^{2} - 16 t + 3 = 0$ $\Rightarrow t = - 8.123, - \frac{1}{3}, + 0.123, + 3.0$ ${\begin{cases} \frac{x + 4}{x - 4} = 66 \Rightarrow \frac{x}{4} = \frac{67}{65} \Rightarrow x = 4.123 \\ \frac{x + 4}{x - 4} = \frac{1}{9} \Rightarrow \frac{x}{4} = - \frac{10}{8} \Rightarrow x = - 5.0 \\ \frac{x + 4}{x - 4} = 0.015 \Rightarrow \frac{x}{4} = \frac{1.015}{0.985} \Rightarrow x = - 4.123 \\ \frac{x + 4}{x - 4} = 9 \Rightarrow \frac{x}{4} = \frac{10}{8} \Rightarrow x = + 5.0 \end{cases}$
	Answered by mind is power last updated on 24/Dec/19

	$\Rightarrow 1 - \frac{2}{\sqrt{(x + 4) (x - 4)}} = \frac{3}{(x - 4) (x + 4)}, x > 0 \neq 4$ $x < 0, x \neq - 4$ $\Rightarrow 1 + \frac{2}{\sqrt{(x - 4) (x + 4)}} = \frac{3}{(4 + x) (x - 4)}, Z = \frac{1}{\sqrt{(x - 4) (x + 4)}}$ $Quadratique$
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