In mathematics, the conjugate of an expression of the form
a + b
is
a − b
{\displaystyle a-b{\sqrt {d}},}
provided that
One says also that the two expressions are conjugate.
In particular, the two solutions of a quadratic equation are conjugate, as per the
in the quadratic formula
Complex conjugation is the special case where the square root is
the imaginary unit.
{\displaystyle (a+b{\sqrt {d}})(a-b{\sqrt {d}})=a^{2}-b^{2}d}
{\displaystyle (a+b{\sqrt {d}})+(a-b{\sqrt {d}})=2a,}
the sum and the product of conjugate expressions do not involve the square root anymore.
This property is used for removing a square root from a denominator, by multiplying the numerator and the denominator of a fraction by the conjugate of the denominator (see Rationalisation).
An example of this usage is:
{\displaystyle {\frac {a+b{\sqrt {d}}}{x+y{\sqrt {d}}}}={\frac {(a+b{\sqrt {d}})(x-y{\sqrt {d}})}{(x+y{\sqrt {d}})(x-y{\sqrt {d}})}}={\frac {ax-dby+(xb-ay){\sqrt {d}}}{x^{2}-y^{2}d}}.}
{\displaystyle {\frac {1}{a+b{\sqrt {d}}}}={\frac {a-b{\sqrt {d}}}{a^{2}-db^{2}}}.}
A corollary property is that the subtraction: leaves only a term containing the root.