4(x^3-1)=3(2x^3+8). Estimate answers to the nearest tenth when necessary.

Let's solve the equation step by step:

The given equation is:

\[
4(x^3 - 1) = 3(2x^3 + 8)
\]

### Step 1: Expand both sides of the equation.
Expand the left side:

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

Expand the right side:

\[
3(2x^3 + 8) = 6x^3 + 24
\]

### Step 2: Set up the equation.
Now rewrite the equation with both sides expanded:

\[
4x^3 - 4 = 6x^3 + 24
\]

### Step 3: Bring all terms involving \(x^3\) to one side and constants to the other side.
Subtract \(6x^3\) from both sides:

\[
4x^3 - 6x^3 - 4 = 24
\]

This simplifies to:

\[
-2x^3 - 4 = 24
\]

### Step 4: Isolate the term involving \(x^3\).
Add 4 to both sides:

\[
-2x^3 = 28
\]

### Step 5: Solve for \(x^3\).
Divide both sides by \(-2\):

\[
x^3 = -14
\]

### Step 6: Solve for \(x\).
Take the cube root of both sides:

\[
x = \sqrt[3]{-14}
\]

Using a calculator to estimate the cube root:

\[
x \approx -2.4
\]

Thus, the solution is approximately \(x \approx -2.4\).

Let me know if you'd like further details or clarifications.

---

Here are 5 related questions you might find interesting:
1. How do you find the cube root of a number manually?
2. What other methods can be used to estimate roots of negative numbers?
3. How can you check your solution to a cubic equation?
4. What are the applications of cubic equations in real-world problems?
5. Can cubic equations have more than one solution?

**Tip**: Always check if your solution satisfies the original equation by substituting the value of \(x\) back into the equation.

4(x^3 - 1) = 3(2x^3 + 8). Estimate answers to the nearest tenth when necessary.

Learn how to solve the cubic equation 4(x^3 - 1) = 3(2x^3 + 8) step-by-step. This algebraic problem involves expanding, isolating cubic terms, and estimating cube roots to the nearest tenth. Suitable for students in Grades 10-12.

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