Solve by carders method x^3+6x^2-12x+32=0

Cardano's method (often referred to as Cardano's formula or method) is used to solve cubic equations of the form:

\[
x^3 + ax^2 + bx + c = 0
\]

Given the equation \(x^3 + 6x^2 - 12x + 32 = 0\), the first step is to remove the \(x^2\) term by substituting \(x = y - \frac{a}{3}\), where \(a\) is the coefficient of \(x^2\).

### Step 1: Substitute \(x = y - \frac{6}{3} = y - 2\)

Substituting \(x = y - 2\) into the original equation:

\[
(y - 2)^3 + 6(y - 2)^2 - 12(y - 2) + 32 = 0
\]

### Step 2: Expand the terms

First, expand \((y - 2)^3\):

\[
(y - 2)^3 = y^3 - 6y^2 + 12y - 8
\]

Next, expand \(6(y - 2)^2\):

\[
6(y - 2)^2 = 6(y^2 - 4y + 4) = 6y^2 - 24y + 24
\]

Finally, expand \(-12(y - 2)\):

\[
-12(y - 2) = -12y + 24
\]

### Step 3: Combine the expanded terms

Now, substitute these expanded terms back into the equation:

\[
y^3 - 6y^2 + 12y - 8 + 6y^2 - 24y + 24 - 12y + 24 + 32 = 0
\]

Simplifying the equation by combining like terms:

\[
y^3 - 24y + 72 = 0
\]

### Step 4: Solve the reduced cubic equation

Now, we solve the equation \(y^3 - 24y + 72 = 0\) using Cardano's method.

First, find \(p\) and \(q\) where:

\[
p = -24, \quad q = 72
\]

### Step 5: Apply Cardano's formula

For a general cubic equation of the form \(y^3 + py + q = 0\), the solutions can be found using:

\[
y = \sqrt[3]{-\frac{q}{2} + \sqrt{\left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3}} + \sqrt[3]{-\frac{q}{2} - \sqrt{\left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3}}
\]

Substituting \(p = -24\) and \(q = 72\):

\[
y = \sqrt[3]{-36 + \sqrt{36^2 + (-8)^3}} + \sqrt[3]{-36 - \sqrt{36^2 + (-8)^3}}
\]

Simplify the terms inside the cube roots:

\[
y = \sqrt[3]{-36 + \sqrt{1296 + (-512)}} + \sqrt[3]{-36 - \sqrt{1296 + (-512)}}
\]

\[
y = \sqrt[3]{-36 + \sqrt{784}} + \sqrt[3]{-36 - \sqrt{784}}
\]

\[
y = \sqrt[3]{-36 + 28} + \sqrt[3]{-36 - 28}
\]

\[
y = \sqrt[3]{-8} + \sqrt[3]{-64}
\]

\[
y = -2 + (-4) = -6
\]

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

Recall the substitution \(x = y - 2\). So:

\[
x = -6 - 2 = -8
\]

Thus, the solution to the equation \(x^3 + 6x^2 - 12x + 32 = 0\) is \(x = -8\).

Would you like further details, or do you have any questions?

Here are five related questions:
1. How do you solve a cubic equation using the discriminant?
2. What are the conditions for a cubic equation to have three real roots?
3. Can Cardano's method be used for quartic equations?
4. How does Cardano's method compare with synthetic division for solving cubic equations?
5. What is the significance of the discriminant in the context of cubic equations?

**Tip:** When solving cubic equations, simplifying the equation first by removing the quadratic term often makes the equation easier to solve using Cardano's method.

Learn how to solve the cubic equation x^3 + 6x^2 - 12x + 32 = 0 step-by-step using Cardano's method. This problem covers Cardano's formula, cubic equations, and root finding techniques. Suitable for advanced high school students and above.

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