Question

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Answer 1

Question:

If the roots of the equation x^2 + 2•c•x + a•b = 0

are real and unequal then prove that the equation x^2 - 2•(a+b)•x + (a^2 + b^2 + 2•c^2) = 0 has unreal roots.

Note:

• If A•x^2 + B•x + C = 0 ,is any quadratic equation,

then its discriminant is given by;

D = B^2 - 4•A•C

• If D = 0 , then the given quadratic equation has real and equal roots.

• If D > 0 , then the given quadratic equation has real and distinct roots.

• If D < 0 , then the given quadratic equation has unreal (imaginary) roots.

Solution:

Here ,

It is given that ,

The roots of the equation x^2 + 2•c•x + a•b = 0

are real and unequal.

Thus,

Its discriminant (D1) will be greater than zero.

ie;

=> D1 > 0

=> (2•c)^2 - 4•1•(a•b) > 0

=> 4•c^2 - 4•a•b > 0

=> 4•( c^2 - a•b ) > 0

=> ( c^2 - a•b ) > 0

=> 0 > a•b - c^2

=> a•b - c^2 < 0 --------(1)

Also,

We have another quadratic equation as;

x^2 - 2•(a+b)•x + (a^2 + b^2 + 2•c^2) = 0.

Thus,

Its discriminant (D2) will be given as ;

=> D2 = {-2•(a+b)}^2 - 4•1•(a^2 + b^2 + 2•c^2)

=> D2 = 4•(a+b)^2 - 4•(a^2 + b^2 + 2•c^2)

=> D2 = 4•{(a+b)^2 - (a^2 + b^2 + 2•c^2)}

=> D2 = 4•(a^2 + b^2 + 2•a•b - a^2 - b^2 - 2•c^2)

=> D2 = 4•(2•a•b - 2•c^2)

=> D2 = 4•2•(a•b - c^2)

=> D2 = 8•(a•b - c^2) ----------(2)

Now,

Considering eq-1 , we have;

=> a•b - c^2 < 0

=> 8•(a•b - c^2) < 8•0

=> 8•(a•b - c^2) < 0

=> D2 < 0 { using eq-(2) }

Since,

The discriminant of the quadratic equation

x^2 - 2•(a+b)•x + (a^2 + b^2 + 2•c^2) = 0 is less than zero ,thus it will have unreal (imaginary) roots .

Hence proved.