Question

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

Answer 1

{ (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