The equation is:

\[
x^3 - 3x^2 + 4 = 0
\]

To solve this cubic equation, we'll attempt factoring or use methods like synthetic division or numerical approximation if necessary.

### Step 1: Trial and error (for rational roots)
Using the Rational Root Theorem, possible rational roots are factors of the constant term (±4) divided by factors of the leading coefficient (±1). Thus, the possible rational roots are ±1, ±2, ±4.

We can test these possible roots by substituting them into the equation.

1. **Test \( x = 1 \):**
   \[
   1^3 - 3(1^2) + 4 = 1 - 3 + 4 = 2 \quad (\text{Not a root})
   \]

2. **Test \( x = -1 \):**
   \[
   (-1)^3 - 3(-1)^2 + 4 = -1 - 3 + 4 = 0 \quad (\text{Root found!})
   \]

So, \( x = -1 \) is a root.

### Step 2: Factor the cubic equation
Now that we know \( x = -1 \) is a root, we can factor the cubic equation as:

\[
x^3 - 3x^2 + 4 = (x + 1)(\text{quadratic factor})
\]

To find the quadratic factor, we perform synthetic division of \( x^3 - 3x^2 + 4 \) by \( x + 1 \).

### Step 3: Synthetic division
Perform synthetic division of \( x^3 - 3x^2 + 4 \) by \( x + 1 \):

\[
\begin{array}{r|rrr}
-1 & 1 & -3 & 0 & 4 \\
   &   & -1 & 4 & -4 \\
\hline
   & 1 & -4 & 4 & 0 \\
\end{array}
\]

The quotient is \( x^2 - 4x + 4 \), so:

\[
x^3 - 3x^2 + 4 = (x + 1)(x^2 - 4x + 4)
\]

### Step 4: Solve the quadratic factor
Now solve the quadratic equation \( x^2 - 4x + 4 = 0 \):

\[
x^2 - 4x + 4 = (x - 2)^2 = 0
\]

Thus, \( x = 2 \) is a repeated root.

### Final solution:
The roots of the equation are:

\[
x = -1, \quad x = 2 \quad (\text{double root})
\]

So, the solutions to the equation \( x^3 - 3x^2 + 4 = 0 \) are:

\[
x = -1, \quad x = 2
\]

Would you like any further details or explanations on this?

---

### Related questions:
1. How is synthetic division used in solving cubic equations?
2. What is the Rational Root Theorem, and how is it applied?
3. How do you factor cubic equations by trial and error?
4. What are the properties of repeated roots in quadratic equations?
5. How do you solve higher-degree polynomials?

### Tip:
Always check possible rational roots using the Rational Root Theorem before attempting synthetic division!

This page explains how to solve the cubic equation x³ - 3x² + 4 using synthetic division and the Rational Root Theorem. Learn step-by-step how to factor the equation and find its roots. Suitable for Grades 10-12.

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