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]