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Solved Example #13

Wednesday, January 22nd, 2014

Question : a, b and c are the sides of a triangle. Equations ax^2 + bx + c = 0 and 3x^2 + 4x + 5 = 0 have a common root. Then angle C is equal to??

Answer : Roots of 3x^2 + 4x + 5 = 0 are complex i.e. they are of the form (p + iq) & (p – iq) where i is iota = sqrt(-1)

For a quadratic equation, complex roots occur in conjugate pairs if the coefficients are real.

In the equation, ax^2 + bx + c = 0, the coefficients are sides of a triangle and hence real. So, if one of roots is common with the other equation say (p+ iq) then the other root will also be compulsorily common. This implies that both equations have the same roots.

This implies that ax^2 + bx + c = 0 , is nothing else but the same equation

i.e. 3x^2 + 4x + 5 = 0

Now, we know that the sides of the triangle are 3,4 & 5 where c = 5.

3,4 & 5 form a Pythagorean triplet making the triangle a right angled triangle.

 The angle opposite to the biggest side is 90 degrees in a right angled triangle, which in this case is C.

Hence, C = 90 degrees.