(3/(x−3))+(5/(x−5))+(7/(x−17))+((19)/(x−19))=x^2 −11x−4


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Question Number 194915 by York12 last updated on 19/Jul/23

$\frac{3}{x - 3} + \frac{5}{x - 5} + \frac{7}{x - 17} + \frac{19}{x - 19} = x^{2} - 11 x - 4$
Commented by BaliramKumar last updated on 19/Jul/23

$typo 7 \to 17$
Commented by York12 last updated on 19/Jul/23

$n o s i r, I a m s u r e$
Commented by BaliramKumar last updated on 19/Jul/23

$i guess just$
Commented by justenspi last updated on 19/Jul/23

$8 s y b e i t i s a t y p o i n m y b o o k$
	Answered by Rasheed.Sindhi last updated on 19/Jul/23
	$(3/(x−3))+(5/(x−5))+((17)/(x−17))+((19)/(x−19))=x^2 −11x−4 (3/(x−3))+1+(5/(x−5))+1+((17)/(x−17))+1+((19)/(x−19))+1=x^2 −11x (x/(x−3))+(x/(x−5))+(x/(x−17))+(x/(x−19))−x^2 +11x=0 x((1/(x−3))+(1/(x−5))+(1/(x−17))+(1/(x−19))−x+11)=0 { ((x=0 ✓)),(((1/(x−3))+(1/(x−19))+(1/(x−5))+(1/(x−17))−x+11=0)) :} ((2x−22)/(x^2 −22x+57))+((2x−22)/(x^2 −22x+85))−(x−11)=0 ((2(x−11))/(x^2 −22x+57))+((2(x−11))/(x^2 −22x+85))−(x−11)=0 (x−11)((2/(x^2 −22x+57))+(2/(x^2 −22x+85))−1)=0 { ((x−11=0⇒x=11✓)),(((2/(x^2 −22x+57))+(2/(x^2 −22x+85))−1=0)) :} (2/(x^2 −22x+57))+(2/(x^2 −22x+85))=1 Let x^2 −22x=y (2/(y+57))+(2/(y+85))=1 2y+170+2y+114=y^2 +142y+4845 y^2 +138y+4561=0 y=−69±10(√2) x^2 −22x=−69±10(√2) ...... ...........$
	$\frac{3}{x - 3} + \frac{5}{x - 5} + \frac{17}{x - 17} + \frac{19}{x - 19} = x^{2} - 11 x - 4$ $\frac{3}{x - 3} + 1 + \frac{5}{x - 5} + 1 + \frac{17}{x - 17} + 1 + \frac{19}{x - 19} + 1 = x^{2} - 11 x$ $\frac{x}{x - 3} + \frac{x}{x - 5} + \frac{x}{x - 17} + \frac{x}{x - 19} - x^{2} + 11 x = 0$ $x (\frac{1}{x - 3} + \frac{1}{x - 5} + \frac{1}{x - 17} + \frac{1}{x - 19} - x + 11) = 0$ ${\begin{cases} x = 0 ✓ \\ \frac{1}{x - 3} + \frac{1}{x - 19} + \frac{1}{x - 5} + \frac{1}{x - 17} - x + 11 = 0 \end{cases}$ $\frac{2 x - 22}{x^{2} - 22 x + 57} + \frac{2 x - 22}{x^{2} - 22 x + 85} - (x - 11) = 0$ $\frac{2 (x - 11)}{x^{2} - 22 x + 57} + \frac{2 (x - 11)}{x^{2} - 22 x + 85} - (x - 11) = 0$ $(x - 11) (\frac{2}{x^{2} - 22 x + 57} + \frac{2}{x^{2} - 22 x + 85} - 1) = 0$ ${\begin{cases} x - 11 = 0 \Rightarrow x = 11 ✓ \\ \frac{2}{x^{2} - 22 x + 57} + \frac{2}{x^{2} - 22 x + 85} - 1 = 0 \end{cases}$ $\frac{2}{x^{2} - 22 x + 57} + \frac{2}{x^{2} - 22 x + 85} = 1$ $L e t x^{2} - 22 x = y$ $\frac{2}{y + 57} + \frac{2}{y + 85} = 1$ $2 y + 170 + 2 y + 114 = y^{2} + 142 y + 4845$ $y^{2} + 138 y + 4561 = 0$ $y = - 69 \pm 10 \sqrt{2}$ $x^{2} - 22 x = - 69 \pm 10 \sqrt{2}$ $. . . . . .$ $. . . . . . . . . . .$
	Commented by York12 last updated on 19/Jul/23

	$t b a n k s$
	Commented by BaliramKumar last updated on 19/Jul/23

	$Nice approach$
	Commented by Rasheed.Sindhi last updated on 20/Jul/23

	🙏
	Answered by Rasheed.Sindhi last updated on 20/Jul/23

	$\frac{3}{x - 3} + \frac{19}{x - 19} + \frac{5}{x - 5} + \frac{17}{x - 17} = x^{2} - 11 x - 4$ $\frac{3 x - 57 + 19 x - 57}{x^{2} - 22 x + 57} + \frac{5 x - 85 + 17 x - 85}{x^{2} - 22 x + 85} = x^{2} - 11 x - 4$ $\frac{22 x - 114}{x^{2} - 22 x + 57} + 2 + \frac{22 x - 170}{x^{2} - 22 x + 85} + 2 = x^{2} - 11 x$ $\frac{2 x^{2} - 22 x}{x^{2} - 22 x + 57} + \frac{2 x^{2} - 22 x}{x^{2} - 22 x + 85} - (x - 11) x = 0$ $x (x - 11) (\frac{2}{x^{2} - 22 x + 57} + \frac{2}{x^{2} - 22 x + 85} - 1) = 0$ $x = 0, x = 11, \frac{2}{x^{2} - 22 x + 57} + \frac{2}{x^{2} - 22 x + 85} - 1 = 0$ $L e t x^{2} - 22 x = y$ $\frac{2}{y + 57} + \frac{2}{y + 85} = 1$ $2 y + 170 + 2 y + 114 = y^{2} + 142 y + 4845$ $y^{2} + 138 y + 4561 = 0$ $y = - 69 \pm 10 \sqrt{2}$ $x^{2} - 22 x = - 69 \pm 10 \sqrt{2}$
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