# Given y=a(x-h)^2+k...what is the correct expression for x: 1. x=sqrt((y+2)/h) 2. x=sqrt((y+2)/h) 3. x=h+-sqrt((y-k)/a) 4. "A simple expression for x cannot be derived."?

Jan 5, 2017

$\text{The third option...........}$

#### Explanation:

We have $y = a {\left(x - h\right)}^{2} + k$

As with any equation, we can divide it, multiply it, add to it, subtract from it etc., provide we do it to BOTH SIDES:

So we want $x$, and we begin be subtracting $k$ from both sides:

$y - k = a {\left(x - h\right)}^{2} + k - k$

$y - k = a {\left(x - h\right)}^{2}$

And now we divide BOTH sides by $a$:

$\frac{y - k}{a} = {\left(x - h\right)}^{2}$

And we take the square root of each side, remembering that a square can have positive or negative roots:

$\sqrt{\frac{y - k}{a}} = \sqrt{{\left(x - h\right)}^{2}}$

$\sqrt{\frac{y - k}{a}} = \pm \left(x - h\right)$

OR $\pm \sqrt{\frac{y - k}{a}} = \left(x - h\right)$

And (finally) $x = h \pm \sqrt{\frac{y - k}{a}}$, as required............