Determine the nature of the roots of the following quadratic equation:

(b + c)x^{2} - (a + b + c)x + a = 0

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#### Solution

The given equation is (b + c)x^{2} - (a + b + c)x + a = 0

The given equation is in the form of ax^{2} + bx + c = 0

where a = (b + c), b = -(a + b + c), c = a

Therefore, the discriminant

D = b^{2} - 4ac

= (-(a + b + c))^{2} - 4 x (b + c) x (a)

= (a + b + c)^{2} - 4ab - 4ac

= a^{2} + b^{2} + c^{2} + 2ab + 2bc + 2ca - 4ab - 4ac

= a^{2} + b^{2} + c^{2} - 2ab + 2bc - 2ac

∵ D > 0

∴ The roots of the given equation are real and distonct.

Concept: Nature of Roots

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