**Either**

$\alpha + \beta = -2k$ | B1B1 | State or use correct results
$\alpha\beta = k$ | M1 | Attempt to find sum of new roots
| A1 | Obtain $4k$
| M1 | Attempt to find product of new roots
| A1 | Obtain $4k$
| B1ft 7 | Correct quadratic equation a.e.f.

$y^2 - 4ky + 4k = 0$

**Or**

$\alpha + \beta = -2k$ | B1 | State or use correct result
$\alpha\beta = k$ | B1 | State or imply form of new roots
$\beta = \frac{-2k}{a}$ | B1 | State correct substitution
$y = \frac{-2k}{x}$ | M1 | Rearrange and substitute for $x$
| A1 | Correct unsimplified equation
| M1 | Attempt to clear fractions
| A1 | Correct quadratic equation a.e.f.

$y^2 - 4ky + 4k = 0$

**Or**

$-k \pm \sqrt{k^2 - k}$ | B1 | Find roots of original equation
$\frac{\alpha + \beta}{\alpha} = \frac{2k}{k + \sqrt{k^2 - k}}$ | B1 | Express both new roots in terms of $k$
$\frac{\alpha + \beta}{\beta} = \frac{2k}{k - \sqrt{k^2 - k}}$ | | 
| M1 | Attempt to find sum of new roots
| A1 | Obtain $4k$
| M1 | Attempt to find product of new roots
| A1 | Obtain $4k$
| B1ft 7 | Correct quadratic equation a.e.f.

$y^2 - 4ky + 4k = 0$

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