The number of real roots of the equation vmatrix 1 & x+1 & x^2+x+1 1 & 2 & x^2+2 3x & 4x-1 & 0 vmatrix = 0 is

Step-by-step solution

Given the equation: vmatrix 1 & x+1 & x^2+x+1 1 & 2 & x^2+2 3x & 4x-1 & 0 vmatrix = 0 Applying the row operation R₁ R₁ - R₂ : vmatrix 0 & x-1 & x-1 1 & 2 & x^2+2 3x & 4x-1 & 0 vmatrix = 0 Applying the column operation C₂ C₂ - C₃ : vmatrix 0 & 0 & x-1 1 & -x^2 & x^2+2 3x & 4x-1 & 0 vmatrix = 0 Expanding the determinant along the first row: (x-1) [1(4x - 1) - (-x^2)(3x)] = 0 (x-1) (3x^3 + 4x - 1) = 0 This gives x = 1 as one real root. For the second factor, let g(x) = 3x^3 + 4x - 1 = 0 . Differentiating with respect