Find the real solutions by factoring.
2 x cubed plus 9 equals x squared plus 18 x

We are tasked with finding the real solutions to the equation:

\[
2x^3 + 9 = x^2 + 18x
\]

### Step 1: Rearrange the equation

First, move all terms to one side of the equation:

\[
2x^3 - x^2 - 18x + 9 = 0
\]

### Step 2: Factor the cubic polynomial

We will try factoring this cubic equation using the method of grouping.

Group terms as follows:

\[
(2x^3 - x^2) - (18x - 9) = 0
\]

Factor each group:

\[
x^2(2x - 1) - 9(2x - 1) = 0
\]

Notice that \( (2x - 1) \) is a common factor. Now factor it out:

\[
(2x - 1)(x^2 - 9) = 0
\]

### Step 3: Factor further

The term \( x^2 - 9 \) is a difference of squares, so it can be factored as:

\[
(2x - 1)(x - 3)(x + 3) = 0
\]

### Step 4: Solve for \( x \)

Now, set each factor equal to zero:

1. \( 2x - 1 = 0 \) ⟹ \( x = \frac{1}{2} \)
2. \( x - 3 = 0 \) ⟹ \( x = 3 \)
3. \( x + 3 = 0 \) ⟹ \( x = -3 \)

### Real Solutions

The real solutions are:

\[
x = \frac{1}{2}, \, x = 3, \, x = -3
\]

Would you like further details or have any questions?

Here are five related questions to explore:
1. How do you factor cubic equations in general?
2. What is the method of grouping in factoring?
3. How do you solve equations involving the difference of squares?
4. What other methods can be used to solve higher-degree polynomials?
5. How do you verify that your solutions to a polynomial are correct?

**Tip**: Always check each solution by substituting it back into the original equation to verify accuracy.

Find the real solutions by factoring. 2x^3 + 9 = x^2 + 18x

Learn how to factor and solve the cubic equation 2x^3 + 9 = x^2 + 18x with step-by-step guidance. This problem covers key algebraic concepts such as factoring by grouping and solving differences of squares. Ideal for students in Grades 10-12.

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