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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:

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

Here, $space space space space space space space space space space straight D space equals space straight b 2 space minus space 4 ac space equals space left parenthesis 1 right parenthesis 2 space minus 4 space open parentheses square root of 2 close parentheses open parentheses square root of 2 close parentheses space equals space 1 space minus space 4 space straight x space 2 space equals space 1 space minus space 8 space equals space minus space 7 space less than space 0$

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

$space space space space space space space space space space space straight x space equals space fraction numerator negative straight b plus-or-minus square root of straight D over denominator 2 straight a end fraction space equals fraction numerator negative straight b space plus-or-minus square root of negative 7 end root over denominator 2 left parenthesis square root of 2 right parenthesis end root end fraction$

i.e. "<pre

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24.

Solve each of the following equations:

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

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25.

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

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