Answer to Question #291929 in Algebra for Rossy

Question

Form the quadratic equation for which the sum of the roots is 5 and the sum of the squares of the roots is 53

Answer 1

Let the two roots be $\alpha$ and $\beta$.

$\alpha +\beta=5\\ \alpha ^2+\beta^2=53$

But,

$(\alpha+\beta)^2=\alpha^2+\beta^2+2\alpha \beta\\ 2\alpha \beta=(\alpha+\beta)^2-\alpha^2-\beta^2\\ 2\alpha \beta=5^2-53=-28\\ \alpha \beta=-14$

Quadratic equation with two roots $\alpha$ and $\beta$ is given as:

$x^2-(\alpha+\beta)x+\alpha \beta=0\\ \text{Substitute in the unknowns, we have the quadratic equation:}\\ x^2-5x-14=0$