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HazzRat

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Can someone smarter than me plz tell me how they did these two steps? If it helps it's for this question:
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Working out
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scaryshark09

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for the first one, i think its meant to be x^2, instead of 2^x

all they do is split up the 2x into x+x and then they rearrange the algebra for the first step
 

scaryshark09

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for the second one they are just factorising, but i think its meant to be xm instead of just x
 

HazzRat

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Does anyone have a cheat sheet for proving shapes in complex numbers? Whenever I'm given a question like "prove these complex points form a parallelogram" I never know how to prove it and the answer's always smthn random like "the diagonals bisect each other". So is there a method of knowing the proof for each shape?
 

Average Boreduser

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Does anyone have a cheat sheet for proving shapes in complex numbers? Whenever I'm given a question like "prove these complex points form a parallelogram" I never know how to prove it and the answer's always smthn random like "the diagonals bisect each other". So is there a method of knowing the proof for each shape?
just look up properties lol. no way of getting around that
 

liamkk112

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Does anyone have a cheat sheet for proving shapes in complex numbers? Whenever I'm given a question like "prove these complex points form a parallelogram" I never know how to prove it and the answer's always smthn random like "the diagonals bisect each other". So is there a method of knowing the proof for each shape?
u just got to memorise the quadrilateral properties no way around it

usually though:
- parallelogram -> pairs of equal side lengths, parallel sides
- square -> equal side lengths, 90 degrees between sides, parallel sides
- rectangle -> 90 degrees between sides, parallel sides
- rhombus -> pairs of equal side lengths, parallel sides, diagonals meet at 90 degrees and bisect

there r also kites but i forget how those work and they're relatively uncommon
 

Luukas.2

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Does anyone have a cheat sheet for proving shapes in complex numbers? Whenever I'm given a question like "prove these complex points form a parallelogram" I never know how to prove it and the answer's always smthn random like "the diagonals bisect each other". So is there a method of knowing the proof for each shape?
There is always a purely algebraic method, which is usually awful. There are sometimes purely geometric methods (like for arg(z - i) = arg(z + 1) etc.). If there isn't an obvious purely geometric approach, the efficient answer is likely to involve:
  • treating the complex numbers as vectors
  • looking for geometric properties that proves the required result
  • demonstrating these properties through algebraic representation of vectors
For example... the complex number z represents a point A in the first quadrant. If O is the origin, B lies in the second quadrant, and OACB is a square, find the complex number representing point C. Under what conditions is C located in the second quadrant.

A diagram should make it obvious that side OB is adjacent to side OA in the square.

Properties of a square then dictate that OB = i.OA, and so the complex number iz represents B.

Then, using vector reasoning:

Hence, the point C is represented by z(1 + i), and so is in the second quadrant if


from the diagram (as A must be in quadrant 1 and, for C to be in quadrant 2 given angle COA is 45 degrees, OA must be inclined at at least 45 degrees above the real axis), or (algebraically), by solving:
 

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