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Complex Numbers and Quadratic Equations

Short Answer Type

21.

solve each of the following equations:

x²+ 3 = 0

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22.

solve each of the following equations:

2x²+ x + 1 = 0

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23.

solve each of the following equations:

$space space space space space space square root of 2 space straight x squared space plus space straight x space plus space square root of 2 space equals space 0$

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24.
Solve each of the following equations:

$space space space space space space space space space space straight x squared space plus space straight x space plus space fraction numerator 1 over denominator square root of 2 end fraction equals space 0$

Here, $space space space space space space space space space space space space straight D space equals space space straight b squared minus 4 ac equals space left parenthesis 1 right parenthesis squared minus 4 left parenthesis 1 right parenthesis open parentheses fraction numerator 1 over denominator square root of 2 end fraction close parentheses equals space 1 space minus fraction numerator 4 over denominator square root of 2 end fraction. fraction numerator square root of 2 over denominator square root of 2 end fraction equals space 1 space minus space 2 space square root of 2 space space less than space 0$

Therefor, roots of given equation are imaginary and are given by

$space space space space space space space space space space space space space straight x space equals space fraction numerator negative straight b space plus-or-minus space square root of straight D over denominator 2 straight a end fraction comma space straight x space equals space fraction numerator negative 1 space plus-or-minus space square root of 1 space minus space 2 square root of 2 end root over denominator 2 left parenthesis 1 right parenthesis end fraction$

i.e. $space space space space space space space space space space straight x space equals space fraction numerator negative 1 space plus space square root of 2 square root of 2 space minus space 1 space straight i end root end root over denominator 2 end fraction comma space fraction numerator negative 1 space minus square root of 2 square root of 2 space minus space 1 space straight i end root end root over denominator 2 end fraction$

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25.

Write the following complex numbers in polar form and hence, find their modulus and argument.

$space space space space space space space space space space space space space 1 half minus fraction numerator square root of 3 over denominator 2 end fraction straight i$