Show that the function $g(x)=sqrt(x-5)-1/(x+3)$ has at least a real root?

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Answer 1 · 2018-04-11T13:42:11

$color(blue)(x=5)$

$sqrt(x-5)-1/(x+3)=0$

Add $1/(x+3)$

$sqrt(x-5)=1/(x+3)$

Squaring:

$x-5=1/(x+3)^2$

Multiply by $(x+3)^2$

$(x+3)^2(x-5)=1$

$(x+3)^2(x-5)-1=0$

Factor:

$(x-5)[(x+3)^2-1/(x-5)]=0$

$x-5=0=>color(blue)(x=5)$

Show that the function g(x)=sqrt(x-5)-1/(x+3)g(x)=sqrt(x-5)-1/(x+3) has at least a real root?