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प्रश्न
Determine the nature of the roots of the following quadratic equation:
(x - 2a)(x - 2b) = 4ab
उत्तर
The given equation is
(x - 2a)(x - 2b) = 4ab
x2 - 2bx - 2ax + 4ab = 4ab
x2 - 2bx - 2ax + 4ab - 4ab = 0
x2 - 2bx - 2ax = 0
x2 - 2(a + b)x = 0
The given equation is of the form of ax2 + bx + c = 0
where a = 1, b = 2(a + b), c = 0
Therefore, the discriminant
D = b2 - 4ac
= (2(a + b))2 - 4 x (1) x (0)
= 4(a + b)2
= 4(a2 + b2 + 2ab)
= 4a2 + 4b2 + 8ab
∵ D > 0
∴ The roots of the given equation are real and distinct.
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