Question

The number of real roots of the equation (x²+2x)²-(x+1)²-55=0

Answer 1

given (x²+2x)²-(x+1)²-55=0 

[(x² +2x +1) -1]² -(x +1)² -55 =0 

(x +1)⁴-2(x +1)² +1 -(x +1)² -55 =0 

(x+1)⁴-3(x +1)² -54 =0 

let (x +1)²= r 

r² -3r -54 =0 

r² -9r +6r -54 =0 

r( r -9) +6(r -9) =0 

r = -6 , 9 

but ( x+1)² ≥ 0 so, (x+1)² ≠ -6 

so, (x +1)² = 9 

x + 1 = ± 3 

x = -1 ±3 

x = -4, and 2 .