Question

Нужно до утра, буду весьма благодарен за любую помощь.

Answer 1

1)
$\frac{1-2i}{1-3i} -(1-2i)(1-3i)+ i^{26}- i^{59} = \\ = \frac{(1-2i)(1+3i)}{(1-3i)(1+3i)} -(1-2i-3i+6 i^{2} )+i^{(24+2)}-i^{(56+3)}= \\ = \frac{1+3i-2i-6i^{2}}{1+3}-(1-5i-6)+ i^{2} - i^{3}= \\ = \frac{7+i}{4} +5+5i-1+i= \\ = \frac{7}{4} +5-1+( \frac{1}{4}+5+1)i=5.75+6.25i$
2)
a)
$m+16n^{2}=( \sqrt{m} +4ni)( \sqrt{m}-4ni)=(4n+ \sqrt{m}i)(4n- \sqrt{m}i)$
b)
$x^{2} +9 y^{2} =(x+3yi)(x-3yi)=(3y+xi)(3y-xi)$
3)
$x^{2} -2x+10=0 \\ x_{12}= \frac{2+- \sqrt{2^{2}-4*10} }{2}=1+- \sqrt{1-10} =1+- \sqrt{-9}=1+- \sqrt{3}i \\ x_{1}=1- \sqrt{3}i ; x_{2}=1+ \sqrt{3}i$