Question

Simplify this expression: x^2+6x+5/x^2-25

Answer 1

`(x^2+6x+5)/(x^2-25)= (x^2+6x+5)/((x-5)(x+5))=(*) \ \ \\ \\.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x -5\\eq 0\ \ and\ \ x+5 \\eq 0\ \ \Rightarrow\ \ x\in R\setminus \{5;-5\}\\ \\x^2+6x+5=(x+5)(x+1)\\ \\because:\\\Delta=6^2-4\cdot1\cdot5=36-20=16\ \ \Rightarrow\ \ √(\Delta) = √(16) =4\\ \\ x_1= (-6-4)/(2\cdot1) = (-10)/(2) =-5,\ \ x_2= (-6+4)/(2\cdot1) = (-2)/(2) =-1\\ \\ \\ (*)= ((x+5)(x+1))/((x-5)(x+5)) = (x+1)/(x-5)`

Answer 2

`(x^2+6x+5)/(x^2-25)=(*);\ x^2-25\\eq0\to x^2\\eq25\to x\\eq-5\ \wedge\ x\\eq5\\\\x^2+6x+5=0\\\\\Delta=6^2-4\cdot1\cdot5=36-20=16;\ \sqrt\Delta=√(16)=4\\\\x_1=(-6-4)/(2\cdot1)=(-10)/(2)=-5;\ x_2=(-6+4)/(2\cdot1)=(-2)/(2)=-1\\\\x^2+6x+5=(x+5)(x+1)`



`(*)=((x+5)(x+1))/((x-5)(x+5))=(x+1)/(x-5)`