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Properties of Determinants

Suppose a,b,c...

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Suppose a,b,c are three real numbers, such that the quadratic equation x²-(a+b+c)x + (ab+bc+ac) = 0 has roots of the form d±ie where d>0 and e≠0 are real numbers [ here i = √-1 ]. Show that (I) the numbers a, b and c are all positive. (II) the numbers √a , √b, √c, form the sides of triangle.

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Q. From the set of first 10 natural numbers, three distinct prime numbers a, b and c are selected to form a quadratic equation ax² + bx + c = 0, having real roots. Find the sum of the roots of all such possible quadratic equations that can be formed.

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a x^{2} + b x + c = 0

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b x^{2} + c x + a = 0

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a = b = c

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+ bx + c = 0 and -

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