Question

Como resolvo o $\sqrt{x+1} -1= \sqrt{x- \sqrt{x+8} }$ o gabarito da 8, mas n chego nesse resultado

Answer 1

$(\sqrt{x+1} -1)^2= (\sqrt{x- \sqrt{x+8} } )^2 \\ \\ \\ ( \sqrt{x+1})^2-2\sqrt{x+1}+1^2= x- \sqrt{x+8} } \\ \\ \\ x+1-2\sqrt{x+1}+1= x- \sqrt{x+8} } \\ \\ \\ x+1-2\sqrt{x+1}+1-x=- \sqrt{x+8} } \\ \\ \\ (2-2\sqrt{x+1})^2=(- \sqrt{x+8} })^2$

$4-8\sqrt{x+1}+ (2\sqrt{x+1} )^2=x+8 \\ \\ \\ 4-8\sqrt{x+1}+4(x+1)=x+8 \\ \\ \\ -8\sqrt{x+1}=x+8-4-4x -4\\ \\ \\ -8\sqrt{x+1}=-3x\\ \\ \\ (-8 \sqrt{x+1})^2= (-3x)^2 \\ \\ \\ 64(x+1) = 9x^2 \\ \\ \\ 64x+64= 9x^2 \\ \\ -9x^2 +64x+64= 0$

$x_{1\ y\ 2}= \dfrac{-b\pm \sqrt{b^2-4ac} }{2a}\qquad \qquad a= -9\qquad b= 64\qquad c= 64 \\ \\ \\ x_{\ 1\ y\ 2}= \dfrac{-64\pm \sqrt{64^2-4(-9)(64)} }{2(-9)} \\ \\ \\ x_{\ 1\ y\ 2}= \dfrac{-64\pm \sqrt{4096+2304} }{-18}\\ \\ \\ x_{\ 1\ y\ 2}= \dfrac{-64\pm \sqrt{6400} }{-18}\\ \\ \\ x_{\ 1\ y\ 2}= \dfrac{-64\pm 80 }{-18} \\ \\ \\ x_{\ 1}= \dfrac{-64+80 }{-18} \qquad \qquad x_{\ 2}= \dfrac{-64- 80 }{-18}$


$x_{\ 1}= \dfrac{16 }{-18} \qquad \qquad x_{\ 2}= \dfrac{-144 }{-18} \\ \\ \\ \boxed{ x_{\ 1}= - \dfrac{8 }{9} } \qquad \qquad \boxed{ x_{\ 2}=8}$

$v\'alido\to x_2= 8\quad \checkmark$

Answer 2

Eu anexei a foto que eu tirei do caderno, é só conferir.