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)#