Question

The conjugate of $3-\sqrt{5+x}$ is $3+\sqrt{5+x}$. Verify

Answer 1

The complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign.

If the conjugate of $(3-\sqrt{5+x})$ is $(3+\sqrt{5+x})$ then the product must be a rational number.

$(3-\sqrt{5+x})(3+\sqrt{5+x})=3^2(\sqrt{5+x}{)}^2=9-(5+x)=9-5-x=4-x(4-x)$ is a rational number.
$\therefore (3+\sqrt{5+x})$ is the conjugate of $(3-\sqrt{5+x})$.