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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:

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

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25.

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

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

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26.

If z₁- -3x + 2iy and z₂ = 6 -5i are equal, find value of x and y.

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27.
If Z₁= (x - 3y) - i (2x + y), and Z₂= -1 - 5i are equal, find values of x a nd y.

Here, Z₁= (x - 3y) -i(2x + y), Z₂= -1 - 5i

Now,    Z₁= Z₂ $rightwards double arrow$ Re(Z₁) = Re (Z₂) $rightwards double arrow$ x-3y $rightwards double arrow$ x = -1                     ...(i)

and          Z₁ =Z₂ $rightwards double arrow$ 1m(Z₁) = 1m(Z₂) $rightwards double arrow$ - 5 = 2x - y $rightwards double arrow$ 2x + y = 5     ...(ii)

Solving (i) and (ii) simultancously, we have

                    x = 2, y = 1