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Question
Determine the nature of the roots of the following quadratic equation:
(b + c)x2 - (a + b + c)x + a = 0
Solution
The given equation is (b + c)x2 - (a + b + c)x + a = 0
The given equation is in the form of ax2 + bx + c = 0
where a = (b + c), b = -(a + b + c), c = a
Therefore, the discriminant
D = b2 - 4ac
= (-(a + b + c))2 - 4 x (b + c) x (a)
= (a + b + c)2 - 4ab - 4ac
= a2 + b2 + c2 + 2ab + 2bc + 2ca - 4ab - 4ac
= a2 + b2 + c2 - 2ab + 2bc - 2ac
∵ D > 0
∴ The roots of the given equation are real and distonct.
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