*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 thatÂ **both equations have the same roots**.

This implies thatÂ **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.

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

Hence,Â **C = 90 degrees**.