To solve the cubic equation \( x^3 - 6x^2 + 11x - 16 = 0 \) using **Cardano's method**, we need to reduce the equation into a depressed cubic form (i.e., without the quadratic term) and apply the method step by step. Here's how we do it:

### Step 1: Depress the cubic equation

A cubic equation of the form:

\[
x^3 + bx^2 + cx + d = 0
\]

can be transformed into a depressed cubic form \( y^3 + py + q = 0 \) by substituting \( x = y + \frac{b}{3} \). In our case, we have:

\[
x^3 - 6x^2 + 11x - 16 = 0
\]

So \( b = -6 \). Let's substitute \( x = y + \frac{-6}{3} = y - 2 \).

### Step 2: Substitute \( x = y - 2 \) into the original equation

We now substitute \( x = y - 2 \) into the cubic equation:

\[
(y - 2)^3 - 6(y - 2)^2 + 11(y - 2) - 16 = 0
\]

Let's expand each term:

- \( (y - 2)^3 = y^3 - 6y^2 + 12y - 8 \)
- \( -6(y - 2)^2 = -6(y^2 - 4y + 4) = -6y^2 + 24y - 24 \)
- \( 11(y - 2) = 11y - 22 \)

Now, putting everything together:

\[
y^3 - 6y^2 + 12y - 8 - 6y^2 + 24y - 24 + 11y - 22 - 16 = 0
\]

Simplify the equation:

\[
y^3 - 12y^2 + 47y - 70 = 0
\]

### Step 3: Solve the depressed cubic equation \( y^3 - 12y^2 + 47y - 70 = 0 \)

Now that the equation is depressed, we would use Cardano's formulas to solve it, but first we should check if there are any rational roots using the Rational Root Theorem.

The Rational Root Theorem suggests possible roots could be factors of -70. Testing these roots (like 2, 5, 7, etc.), we eventually find that:

\[
y = 2
\]

is a root.

### Step 4: Factor the cubic

Now that we know \( y = 2 \) is a root, we can factor the cubic equation as:

\[
(y - 2)(y^2 - 10y + 35) = 0
\]

### Step 5: Solve the quadratic

Now solve the quadratic \( y^2 - 10y + 35 = 0 \) using the quadratic formula:

\[
y = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(1)(35)}}{2(1)}
\]

\[
y = \frac{10 \pm \sqrt{100 - 140}}{2}
\]

\[
y = \frac{10 \pm \sqrt{-40}}{2}
\]

\[
y = \frac{10 \pm 2i\sqrt{10}}{2}
\]

\[
y = 5 \pm i\sqrt{10}
\]

### Step 6: Back-substitute to find \( x \)

Recall that \( x = y - 2 \). So the solutions for \( x \) are:

\[
x_1 = 2 - 2 = 0
\]

\[
x_2 = 5 - 2 + i\sqrt{10} = 3 + i\sqrt{10}
\]

\[
x_3 = 5 - 2 - i\sqrt{10} = 3 - i\sqrt{10}
\]

### Final Solutions:

The solutions to the cubic equation \( x^3 - 6x^2 + 11x - 16 = 0 \) are:

\[
x_1 = 0, \quad x_2 = 3 + i\sqrt{10}, \quad x_3 = 3 - i\sqrt{10}
\]

Would you like more details on any step, or have any further questions?

### 5 Related Questions:
1. What is the general form of Cardano's formula for cubic equations?
2. How do you handle complex roots in Cardano's method?
3. Can all cubic equations be solved by the Cardano method?
4. How does the Rational Root Theorem help in solving polynomials?
5. What is the geometric interpretation of complex roots?

### Tip: 
When solving cubic equations, always check for rational roots using the Rational Root Theorem first. It can help simplify the equation considerably before using more advanced methods like Cardano’s.

X³ - 6x² + 11x - 16 = 0 solve by Cardano's method

Solve the cubic equation X³ - 6x² + 11x - 16 using Cardano's method. This step-by-step solution explains how to depress the cubic equation, apply the Rational Root Theorem, and find the complex and real roots. Perfect for students in Grade 11-12 or college-level math courses.

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