5 Answers. If α and β are the roots of the quadratic equation x2+bx+c=0 ( a is not equal to zero), then equation whose roots are α and β is x2−(α+β)x+αβ=0. The equation whose roots are α β and β α is: x2−(α β+β α)x+(α β)(β α)=0.. Explanation: So since α is a root of x2−x+1=0 we have:. given the roots of equation 3x^2 - 6x +1 =0 are \alpha , \beta , find equation with integer coefficients whose roots are \alpha*\beta^2 and \alpha^2 *\beta. The sum of the new roots is and the product of the new roots is . You already know the values of and . I see you've digged out quite an old thread there.. To ask Unlimited Maths doubts download Doubtnut from - https://goo.gl/9WZjCW if `alpha , beta` are root of `ax^2+bx+c=0` then `(1/alpha^2-1/beta^2)^2`. A problem that involves the use of the sum and product of quadratic roots.. roots/zeros/solutions to the quadratics can be found with the quadratic formula, by factoring, or completing the square. from inspection, the roots will be irrational and therefore not factorable. i will find the roots by completing the square: x^2 + 4x + 1 = 0 (x + 2)^2 = -1 + 4.