Conjugate (square roots)

In mathematics, the conjugate of an expression of the form ${\displaystyle a+b{\sqrt {d}}}$ is ${\displaystyle a-b{\sqrt {d}},}$ provided that ${\displaystyle {\sqrt {d}}}$ does not appear in a and b. One says also that the two expressions are conjugate.

In particular, the two solutions of a quadratic equation are conjugate, as per the ${\displaystyle \pm }$ in the quadratic formula ${\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}}$.

Complex conjugation is the special case where the square root is ${\displaystyle i={\sqrt {-1}},}$ the imaginary unit.