We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website.
Please click on "Accept cookies" if you agree to the setting of cookies. Cookies that do not require consent remain unaffected by this, see
cookie policy and privacy policy.
DECLINE COOKIES

the negative reciprocal of the sum of all values of x that satisfy the equation x-sqrt(x+1)/x+sqrt(x+1)=11/5

Below is my steps but idk if its right

36x2 (x to the power of 2) =162 (16 to the power of 2) (x+1)

and then the 16 to the power of to eventually equals 256x+256

ihatespiritofmath Jan 1, 2019

#1**+1 **

\(\text{I assume this is (parens are your friend)}\\ \dfrac{x-\sqrt{x+1}}{x+\sqrt{x+1}}=\dfrac{11}{5}\)

\(\text{Let's pull a bit of a trick here}\\ a=\sqrt{x+1}\\ \dfrac{x-a}{x+a}=\dfrac{11}{5}\\ 5x-5a=11x+11a\\ 6x+16a=0\\ 3x=-8a = -8\sqrt{x+1}\\\)

\(9x^2 = 64(x+1)\\ 9x^2 - 64x-64 = 0\\ \text{Using the quadratic formula, (which I leave to you)}\\ x = 8,~-\dfrac{9}{8}\\ \text{Since we squared the equation we must check these roots}\)

\(x=8 \Rightarrow a = 3\\ \dfrac{8-3}{8+3} = \dfrac{5}{11} \neq \dfrac{11}{5}\\ \text{8 is a extraneous solution}\\ x=-\dfrac{8}{9}\Rightarrow a = \dfrac{1}{3}\\ \dfrac{-\dfrac{8}{9}-\dfrac 1 3}{-\dfrac{8}{9}+\dfrac 1 3} = \dfrac{11}{5}\\ \text{so }x=-\dfrac{8}{9} \text{ is an actual root, and the only one}\)

\(-\dfrac{1}{-\dfrac 8 9} = \dfrac 9 8\)

.Rom Jan 1, 2019