LAST ONE plz help some 1

There is only one solution for  x  when the discrimant is  0 , that is, when...

$(b+\frac1b)^2-4(1)(c)\ =\ 0\\~\\ (b+\frac1b)^2-4c\ =\ 0\\~\\ (b+\frac1b)(b+\frac1b)-4c\ =\ 0\\~\\ b^2+2+\frac{1}{b^2}-4c\ =\ 0\\~\\ b^2+(2-4c)+\frac{1}{b^2}\ =\ 0\\~\\ b^4+(2-4c)b^2+1\ =\ 0\qquad\text{Let}\qquad u=b^2\\~\\ u^2+(2-4c)u+1\ =\ 0$

There is only one solution for  u, and thus only one positive value of  b ,  when...

$(2-4c)^2-4\ =\ 0\\~\\ (2-4c)^2\ =\ 4\\~\\ 2-4c\ =\ 2\qquad\text{or}\qquad2-4c\ =\ -2\\~\\ c\ =\ 0\phantom{2-4}\qquad\text{or}\qquad c\ =\ 1$

The non-zero value of  c  is  1 .

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To check this answer, let's find the values of  b  for which   $x^2+(b+\frac1b)x+1=0$   has only one solution.

$x^2+(b+\frac1b)x+1=0$     has only one solution when....

$(b+\frac1b)^2-4\ =\ 0\\~\\ b^2-2+\frac{1}{b^2}\ =\ 0\\~\\ b^4-2b^2+1\ =\ 0\\~\\ (b^2-1)^2\ =\ 0\\~\\ b\ =\ \pm1$

There is only one positive value of  b  for which there is one solution to the equation  $x^2+(b+\frac1b)x+1=0$

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The exact same question has been asked here before.

https://web2.0calc.com/questions/help_77926