By completing the square or otherwise, express $x$ in terms of $y$ given that $y = x^2 - 4x + 6$. Answer x = ___ or ___ Solve the equation $38 = a^6 - 4a^3 + 6$. Answer a = ___ or ___

1. Complete the square for the quadratic expression in x.

$y=x^2-4x+6$

$y=(x-2)^2+2$

2. Isolate the squared term.

$(x-2)^2=y-2$

3. Take square roots and solve for x, including both signs.

$x-2=\pm\sqrt{y-2}$

$x=2\pm\sqrt{y-2}$

1. Recognise the expression as a quadratic in $a^3$ and substitute.

Let $x=a^3$. Then
$38=x^2-4x+6.$

2. Use the result from part (i), or solve the quadratic, to find the possible values of $a^3$.

From part (i), $x=2\pm\sqrt{y-2}$.

When $y=38$,
$x=2\pm\sqrt{38-2}=2\pm\sqrt{36}=2\pm6.$
So
$x=8\quad\text{or}\quad x=-4.$

3. Convert the solutions for $x$ back to values of $a$.

Since $x=a^3$,
$a^3=8\quad\text{or}\quad a^3=-4.$
Hence
$a=2\quad\text{or}\quad a=-\sqrt[3]{4}.$