$k = x^{\frac{1}{4}}$ | M1* | Use a substitution to obtain a quadratic or factorise into 2 brackets each containing $x^{\frac{1}{4}}$ | No marks unless evidence of substitution (quadratic rooting or squaring of roots found). = 0 may be implied. | Allow $x = x^{\frac{1}{4}}$ as a substitution.

$3k^2 - 8k + 4 = 0$ | DM1 | Correct method to solve quadratic |

$(3k-2)(k-2) = 0$ | | |

$k = \frac{2}{3}$ or $k = 2$ | A1 | | No marks if straight to quadratic formula to get $x = "2"$ $x = "2"$ and no further working | No marks if $k = x^{\frac{1}{4}}$ then $3k - 8k + 4 = 0$

$x = (\frac{2}{3})^4$ or $x = 2^4$ | M1 | Attempt to calculate $k^{\frac{1}{4}}$ |

$x = \frac{16}{81}$ or $x = 16$ | A1, 5, 5 | | SC If M0 Spotted solutions www | B1 each Justifies 2 solutions exactly B3

If candidates use $k = x^{\frac{1}{4}}$ and rearrange: | | |

$3k - 8\sqrt{k} + 4 = 0$ | M1* | Substitute, rearrange and square both sides | |

$8\sqrt{k} = 3k + 4$ | | |

$64k = 9k^2 + 24k + 16$ | M1* | Substitute, rearrange and square both sides | |

$9k^2 - 40k + 16 = 0$ | | |

$(9k-4)(k-4)=0$ | DM1 | Correct method to solve quadratic | |

$k = \frac{4}{9}$ or $k = 4$ | A1 | |

$x = (\frac{4}{9})^2$ or $x = 4^2$ | A1 | |

$x = \frac{16}{81}$ or $x = 16$ | A1 | |