69
If $\alpha$ and $\beta$ are roots of the quadratic equation $x^2 - 7x + 10 = 0$, find the quadratic equation whose roots are $\alpha^2$ and $\beta^2$.
Show SolutionHide Solution↓
$x^2 - 7x + 10 = 0$
$\alpha + \beta = 7, \alpha\beta = 10$
$\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = 49 - 20 = 29$
$\alpha^2\beta^2 = (10)^2 = 100$
Quadratic Equation with roots $\alpha^2, \beta^2$ is
$\therefore x^2 - (\alpha^2 + \beta^2)x + \alpha^2\beta^2 = 0$
i.e. $x^2 - 29x + 100 = 0$
$\alpha + \beta = 7, \alpha\beta = 10$
$\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = 49 - 20 = 29$
$\alpha^2\beta^2 = (10)^2 = 100$
Quadratic Equation with roots $\alpha^2, \beta^2$ is
$\therefore x^2 - (\alpha^2 + \beta^2)x + \alpha^2\beta^2 = 0$
i.e. $x^2 - 29x + 100 = 0$