How can I solve for z? 3+sqrt(z-6)=sqrt(z+9)
This is going to be a problem in two steps. The base example here is how to solve the following for z:
$\displaystyle \sqrt{z + 1} + 2 = 3$
The general rule is to isolate the square root, then square both sides. In your problem we need to do this twice.
So
$\displaystyle 3 + \sqrt{z-6} = \sqrt{z+9}$
One square root is already isolated, so square both sides:
$\displaystyle 9 + 6 \sqrt{z - 6} + (z - 6) = z + 9$
Now isolate the square root:
$\displaystyle 6 \sqrt{z - 6} = 6$
And square both sides.
Always always always when you get a final answer for z make sure it is a solution to the original equation. Extra solutions tend to come out that don't satisfy the original equation.
-Dan
$\displaystyle (3 + \sqrt{z - 6})^3 = 3^2 + (2)(3)(\sqrt{z - 6}) + (\sqrt{z - 6})^2 = 9 + 6\sqrt{z - 6} + (z - 6)$
because:
$\displaystyle (a + b)^2 = (a + b)(a + b) = a(a + b) + b(a + b) = a^2 + ab + ba + b^2 = a^2 + 2ab + b^2$