Q.Express $\dfrac{1+2i}{3-i}+\dfrac{1-2i}{3+i}$ in the form $a+ib$ and show that it is a real number.

The denominators $3-i$ and $3+i$ are conjugates, so their product is real: $(3-i)(3+i)=9+1=10$. Combine the two fractions over this common denominator: numerator $=(1+2i)(3+i)+(1-2i)(3-i)$. Expand each: $(1+2i)(3+i)=3+i+6i+2i^2=3+7i-2=1+7i$; $(1-2i)(3-i)=3-i-6i+2i^2=3-7i-2=1-7i$. Adding: $(1+7i)+(1-7i)=2+0i=2$. So the sum i …

Combine the two fractions over the common denominator $(3-i)(3+i)$, which is automatically real since the two denominators are conjugates; expan …

• Expanding $(1+2i)(3+i)$ and $(1-2i)(3-i)$ with a sign error, which would leave a nonzero imaginary part and wrongly suggest the sum is not real …