Question

Determine the nature of the roots (x-2a)(x-2b)=4ab

Answer 1

$(x - 2a)(x - 2b) = 4ab \\ {x}^{2} - 2bx - 2ax + 4ab = 4ab \\ {x}^{2} - 2x(b + a) = 0 \\ x(x - 2(b + a)) = 0$
Either root is 0 or 2(b+a).

by nature one is both are positive because 0 is always taken as positive integer.

Answer 2

given equation is

(x-2a)(x-2b)=4ab

x²-2bx-2ax-4ab=0

x²+[-2(a+b)]x+(-4ab)=0

here A=0;B=-2(a+b);C=-4ab

determinent=B²-4AC

=[-2(a+b)]²-4(0)(-4ab)

=4(a+b)²-0

=[2(a+b)]²

we know that any x€R;x²>0

so [2(a+b)]²>0

determinent>0

so,roots of given equation are positive integers.

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hope it'll help you