Basic Square Root Help

**Espionage** · January 8th 2013, 05:09 PM

How can I solve for z? 3+sqrt(z-6)=sqrt(z+9)

**topsquark** · January 8th 2013, 05:31 PM

Originally Posted by Espionage

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:
$\sqrt{z + 1} + 2 = 3$

The general rule is to isolate the square root, then square both sides. In your problem we need to do this twice.

So
$3 + \sqrt{z-6} = \sqrt{z+9}$

One square root is already isolated, so square both sides:
$9 + 6 \sqrt{z - 6} + (z - 6) = z + 9$

Now isolate the square root:
$6 \sqrt{z - 6} = 6$

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

**Espionage** · January 8th 2013, 05:41 PM

Where does the 6 come from when you square both sides the first time?

**Deveno** · January 8th 2013, 06:54 PM

$(3 + \sqrt{z - 6})^3 = 3^2 + (2)(3)(\sqrt{z - 6}) + (\sqrt{z - 6})^2 = 9 + 6\sqrt{z - 6} + (z - 6)$

because:

$(a + b)^2 = (a + b)(a + b) = a(a + b) + b(a + b) = a^2 + ab + ba + b^2 = a^2 + 2ab + b^2$

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