Question

The number of real roots of the equation $(x^2+2x{)}^2-(x+1{)}^2-55=0$

• A. 2
• B. 1
• C. 4
• D. none of these

Answer 1

The correct option is A

2

$(x^2+2x{)}^2-(x+1{)}^2-55=0\Rightarrow (x^2+2x+1-1{)}^2-(x+1{)}^2-55=0\Rightarrow {(x+1{)}^2-1}^2-(x+1{)}^2-55=0\Rightarrow {(x+1{)}^2}^2+1-3(x+1{)}^2-55=0\Rightarrow {(x+1{)}^2}^2-3(x+1{)}^2-54=0Letp=(x+1{)}^2\Rightarrow p^2-3p-54=0\Rightarrow p^2-9p+6p-54=0\Rightarrow (p+6)(p-9)=0\Rightarrow (p+6)(p-9)=0\Rightarrow p=9orp=-6Rejectingp=-6\Rightarrow (x+1{)}^2=9\Rightarrow x^2+2x-8=0\Rightarrow x^2+4x-2x-8=0\Rightarrow (x+4\left)\right(x-2)=0\Rightarrow x=2,x=-4$