We need to solve the equation $123 + (3x - 9)^2 = 323$ for $x$.

1. Problem Description

We need to solve the equation $123 + (3x - 9)^2 = 323$ for $x$.

2. Solution Steps

First, we isolate the squared term by subtracting 123 from both sides of the equation:

$123 + (3x - 9)^2 - 123 = 323 - 123$

$(3x - 9)^2 = 200$

Next, we take the square root of both sides:

$\sqrt{(3x - 9)^2} = \pm \sqrt{200}$

$3x - 9 = \pm \sqrt{200}$

We simplify $\sqrt{200}$:

$\sqrt{200} = \sqrt{100 \cdot 2} = \sqrt{100} \cdot \sqrt{2} = 10\sqrt{2}$

So we have

$3x - 9 = \pm 10\sqrt{2}$

Now, we isolate $x$ by adding 9 to both sides:

$3x = 9 \pm 10\sqrt{2}$

Finally, we divide both sides by 3:

$x = \frac{9 \pm 10\sqrt{2}}{3}$

$x = 3 \pm \frac{10\sqrt{2}}{3}$

3. Final Answer

The solutions are $x = 3 + \frac{10\sqrt{2}}{3}$ and $x = 3 - \frac{10\sqrt{2}}{3}$.

$x = 3 \pm \frac{10\sqrt{2}}{3}$