The roots of a quadratic equation $2{x^2} + 4x - 1 = 0$ are $\alpha$ and $\beta$.

Without solving the equation,

(a)     find the value of ${\alpha ^2} + {\beta ^2}$;

(b)     find a quadratic equation with roots ${\alpha ^2}$ and ${\beta ^2}$.

(a) using the formulae for the sum and product of roots:

$\alpha  + \beta  =  - 2$     A1

$\alpha \beta  =  - \frac{1}{2}$     A1

${\alpha ^2} + {\beta ^2} = {(\alpha  + \beta )^2} - 2\alpha \beta$     M1

$= {( - 2)^2} - 2\left( { - \frac{1}{2}} \right)$

$= 5$     A1

Note:     Award M0 for attempt to solve quadratic equation.

[4 marks]

(b)     $(x - {\alpha ^2})(x - {\beta ^2}) = {x^2} - ({\alpha ^2} + {\beta ^2})x + {\alpha ^2}{\beta ^2}$     M1

${x^2} - 5x + {\left( { - \frac{1}{2}} \right)^2} = 0$     A1

${x^2} - 5x + \frac{1}{4} = 0$

Note:     Final answer must be an equation. Accept alternative correct forms.

[2 marks]

Total [6 marks]