Question

URGENT!!! PLEASE help me with this question!

Answer 1

`\huge\boxed{x = 8,\ but\ x = 3\ is\ extraneous}`

Explanation:

`√(x+1)+5=x\\\\\text{Domain:}\ x+1\geq0\to x\geq-1\\\\√(x+1)+5=x\qquad\text{subtract 5 from both sides}\\√(x+1)=x-5\qquad\text{square both sides}\\(√(x+1))^2=(x-5)^2\qquad\text{use}\ (√(x))^2=x\ \text{and}\ (a-b)^2=a^2-2ab+b^2\\x+1=x^2-2(x)(5)+5^2\\x+1=x^2-10x+25\qquad\text{subtract}\ x\ \text{and}\ 1\ \text{from both sides}\\0=x^2-11x+24`

`ab=24;\ a+b=-11\\a=-8;\ b=-3\\\\x^2-11x+24=0\\x^2-8x-3x+24=0\qquad\text{distribute}\\x(x-8)-3(x-8)=0\qquad\text{distribute}\\(x-8)(x-3)=0\\\\\text{The product is 0 if one of the factors is 0. Therefore}\\(x-8)(x-3)=0\iff x-8=0\ \vee\ x-3=0\\x-8=0\qquad\text{add 8 to both sides}\\\boxed{x=8}\in D\\x-3=0\qquad\text{add 3 to both sides}\\\boxed{x=3}\in D`

`\text{For}\ x=8:\\√(8+1)+5=8\\√(9)+5=8\\3+5=8\\8=8\qquad\bold{CORRECT}\\\text{For}\ x=3:\\√(3+1)+5=3\\\sqrt4+5=3\\2+5=3\\7=3\qquad\bold{FALSE}`