To solve the equation \(1 + 8x^3 = 1 + 2x\), let's follow these steps:

### Step 1: Simplify the Equation
Subtract 1 from both sides of the equation to eliminate the constant on the left side:
\[
8x^3 = 2x
\]

### Step 2: Factor the Equation
We can factor out \(x\) from both terms on the right side:
\[
8x^3 - 2x = 0
\]
\[
x(8x^2 - 2) = 0
\]

### Step 3: Solve for \(x\)
Now, set each factor equal to zero:
1. \(x = 0\)
2. \(8x^2 - 2 = 0\)

For the second factor, solve for \(x\):
\[
8x^2 = 2
\]
\[
x^2 = \frac{2}{8} = \frac{1}{4}
\]
\[
x = \pm \sqrt{\frac{1}{4}} = \pm \frac{1}{2}
\]

### Final Answer:
The solutions to the equation \(1 + 8x^3 = 1 + 2x\) are:
\[
x = 0, \quad x = \frac{1}{2}, \quad x = -\frac{1}{2}
\]

Would you like more details on this solution, or do you have any questions?

### Follow-up Questions:
1. How do you factor polynomials?
2. Can you solve \(x^2 - 4x + 4 = 0\)?
3. What are the steps to solve quadratic equations?
4. Can you solve \(2x^2 + 5x - 3 = 0\)?
5. How do you solve cubic equations in general?
6. What are the different methods for solving equations?
7. Can you explain the difference between a root and a factor?
8. What is the quadratic formula?

**Tip:** When solving polynomial equations, always look for common factors first to simplify the problem.

Learn how to solve the equation 1 + 8x^3 = 1 + 2x with detailed step-by-step instructions. This problem involves algebraic manipulation, factoring polynomials, and solving cubic equations. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation