1. Simplify the expression (3 - 4i)(1 + 5i) - (2 - i)

Mathematics

1 answer:

lord [1]3 years ago

5 0

Step-by-step explanation:

(3 − 4i) (1 + 5i) − (2 − i)

3 + 15i − 4i − 20i² − 2 + i

1 + 12i + 20

21 + 12i

(a² + 3a − 7) (2a² − a + 4)

a² (2a² − a + 4) + 3a (2a² − a + 4) − 7 (2a² − a + 4)

2a⁴ − a³ + 4a² + 6a³ − 3a² + 12a − 14a² + 7a − 28

2a⁴ + 5a³ − 13a² + 19a − 28

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Find the coordinates of the point three tenths of the way from A(-4, -8) to B(11, 7)

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let's first off take a peek at those values.

let's say the point with those coordinates is point C, so C is 3/10 of the way from A to B.

meaning, we take the segment AB and cut it in 10 equal pieces, AC takes 3 pieces, and CB takes 7 pieces, namely AC and CB are at a 3:7 ratio.

$\bf ~~~~~~~~~~~~\textit{internal division of a line segment}
\\\\\\
A(-4,-8)\qquad B(11,7)\qquad
\qquad \stackrel{\textit{ratio from A to B}}{3:7}
\\\\\\
\cfrac{A\underline{C}}{\underline{C} B} = \cfrac{3}{7}\implies \cfrac{A}{B} = \cfrac{3}{7}\implies 7A=3B\implies 7(-4,-8)=3(11,7)\\\\[-0.35em]
~\dotfill\\\\
C=\left(\frac{\textit{sum of "x" values}}{\textit{sum of ratios}}\quad ,\quad \frac{\textit{sum of "y" values}}{\textit{sum of ratios}}\right)\\\\[-0.35em]
~\dotfill$

$\bf C=\left(\cfrac{(7\cdot -4)+(3\cdot 11)}{3+7}\quad ,\quad \cfrac{(7\cdot -8)+(3\cdot 7)}{3+7}\right)
\\\\\\
C=\left( \cfrac{-28+33}{10}~~,~~\cfrac{-56+21}{10} \right)\implies C=\left( \cfrac{5}{10}~~,~~\cfrac{-35}{10} \right)
\\\\[-0.35em]
\rule{34em}{0.25pt}\\\\
~\hfill C=\left( \frac{1}{2}~,~-\frac{7}{2} \right)~\hfill$