The Conjugate

The conjugate of $a+bi$ is $a-bi$: the same real part, with the sign of the imaginary part flipped. The conjugate of $2+7i$ is $2-7i$, and the conjugate of $6-i$ is $6+i$.

Multiply a number by its conjugate: $\left(2+7i\right)\left(2-7i\right) = 4-14i+14i-49{i}^{2}$. The two i terms are opposites and cancel, and $-49{i}^{2}$ is +49. The result is 4 + 49, which is 53, a real number.

For real a and b that always happens: $\left(a+bi\right)\left(a-bi\right) = {a}^{2}+{b}^{2}$. Compare $\left(a+b\right)\left(a-b\right) = {a}^{2}-{b}^{2}$ from Algebra: the minus turns into a plus because ${i}^{2} = -1$. This is the tool for the next lesson, since multiplying by the conjugate turns a complex number into a real one.