2010 Paper 1 Q9 - roots of quadratic equation

Q:  Given that the roots of 3x² – 2x + 1 = 0 are α and β, find the quadratic equation whose roots are α+2β and 2α+β

Ahhh…roots of the equation… I love quadratic!

OK, to answer this type of question, we must first recall that ANY quadratic equation can be defined by the sum and products of its roots as such:

                x² – (sum of roots)x + product of roots = 0

This is a GIVEN. One must always notice that the quadratic equation is given by x² and not ax².  Hence, in this question, given that the quadratic equation begins with 3x², it means that we have to divide throughout by 3 first. This gives us

Hence,

α+β = 2/3   and   αβ = 1/3

If there is another quadratic equation with roots α+2β and 2α+β, then, the sum of the roots are

α+2β + 2α+β = 3α+3β = 3(α+β) = 2

The product of the roots will be (α+2β)(2α+β)

                                                = 2α² + 5αβ + 2β²

                                                = 2(α+β)² + αβ

                                                = 2(4/9) + 1/3

                                                = 8/9 + 3/9

                                                = 11/9

Hence, the quadratic equation with roots of α+2β and 2α+β is