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The given equation is:

\[
x^2 = 3(2x + 9)
\]

Let's solve this step by step.

### Step 1: Expand the right-hand side
We distribute the 3 on the right-hand side:

\[
x^2 = 3(2x + 9) = 6x + 27
\]

So the equation becomes:

\[
x^2 = 6x + 27
\]

### Step 2: Rearrange the equation
Move all terms to one side of the equation:

\[
x^2 - 6x - 27 = 0
\]

### Step 3: Solve the quadratic equation
We now have the quadratic equation:

\[
x^2 - 6x - 27 = 0
\]

We can solve this using the quadratic formula, which is given by:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \(x^2 - 6x - 27 = 0\), the coefficients are:
- \(a = 1\)
- \(b = -6\)
- \(c = -27\)

Now, substitute the values into the quadratic formula:

\[
x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(-27)}}{2(1)}
\]
\[
x = \frac{6 \pm \sqrt{36 + 108}}{2}
\]
\[
x = \frac{6 \pm \sqrt{144}}{2}
\]
\[
x = \frac{6 \pm 12}{2}
\]

### Step 4: Calculate the two possible solutions

1. For \(x = \frac{6 + 12}{2} = \frac{18}{2} = 9\)
2. For \(x = \frac{6 - 12}{2} = \frac{-6}{2} = -3\)

### Final Answer
The truth set (solution set) of the equation is:

\[
\{x = 9, x = -3\}
\]

Would you like further details or explanations on any part of this solution?

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Here are 5 related questions to expand your understanding:

1. What are the possible methods for solving quadratic equations, and when is each method appropriate?
2. Can all quadratic equations be solved using the quadratic formula?
3. How would the solution change if the right-hand side of the equation were different?
4. What is the significance of the discriminant in the quadratic formula?
5. How can you verify if the solutions you found are correct?

**Tip:** Always check your solutions by substituting them back into the original equation to ensure they satisfy it.

Find the truth set of the equation x^2 = 3(2x + 9).

Solve the quadratic equation x^2 = 3(2x + 9) with a detailed step-by-step approach. This solution involves expanding, rearranging, and using the quadratic formula. Suitable for students in Grades 8-10.

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