quadratic equation, calculate, the sum of the roots = α β + = − = and the product of the roots = αβ β = − + = − . Hence a quadratic equation with required roots is x = . Multiplying this equation by , gives x = which is also a quadratic equation having roots α + 2and β + . Example .
If α and β are the roots of the quadratic equation x , construct a quadratic equation whose roots are α and β . Since α and β are the roots of the quadratic equation, we have α β = and αβ = . Thus, to construct a new quadratic equation, Sum of the roots = α β β αβ = − Product of the roots = α β αβ = ( ) = Thus a required quadratic equation is x . From this we see that x = is a quadratic equation with roots α and β .
Remark In Examples . and . , we have computed the sum and the product of the roots using the known α β and αβ . In this way we can construct quadratic equation with desired roots, provided the sum and the product of the roots of a new quadratic equation can be written using the sum and the product of the roots of the given quadratic equation.
We note that we have not solved the given equation; we do not know the values of α and β even after completing the task. - -