The equation $x^{3} - 3x + q = 0$ will have two roots equal, if the value of $q$ is
- $ \pm 2$
- $ \pm 1$
- $ \pm 3$
- $ \pm 4$
For x^3 - 3x + q = 0 to have equal roots, the derivative 3x^2 - 3 must be zero at the root. So x^2 = 1, x = 1 or -1. If x = 1, 1 - 3 + q = 0 => q = 2. If x = -1, -1 + 3 + q = 0 => q = -2. Thus q = +/- 2.
For the cubic equation x cubed minus 3x plus q equals 0 to have two equal roots, its discriminant must equal zero. The formula for the discriminant of a cubic is 18abcd minus 4b cubed d plus b squared c squared minus 4ac cubed minus 27a squared d squared, which here simplifies to 4q cubed minus 27q squared. Setting this to zero gives q equals 0 or q equals plus or minus 2; q equals 0 results in three real, distinct roots. Therefore, the required value is plus or minus 2.