Find the roots with solutions x square root 5 - 5x square root 4 + 6x square root 3 - x = 0

We are given the equation:

\[
x \sqrt{5} - 5x \sqrt{4} + 6x \sqrt{3} - x = 0
\]

### Step-by-step solution:

1. **Simplify the terms involving square roots**:
   - \(\sqrt{5}\) remains as \(\sqrt{5}\).
   - \(\sqrt{4} = 2\), so the second term becomes \(5x \cdot 2 = 10x\).
   - \(\sqrt{3}\) remains as \(\sqrt{3}\).

   The equation becomes:
   \[
   x \sqrt{5} - 10x + 6x \sqrt{3} - x = 0
   \]

2. **Combine like terms**:
   Group the terms involving \(x\):
   \[
   (x \sqrt{5} + 6x \sqrt{3} - 10x - x) = 0
   \]
   Simplifying the constants:
   \[
   (x \sqrt{5} + 6x \sqrt{3} - 11x) = 0
   \]

3. **Factor out \(x\)**:
   \[
   x (\sqrt{5} + 6 \sqrt{3} - 11) = 0
   \]

4. **Solve for \(x\)**:
   For the product to be zero, either \(x = 0\) or the expression in parentheses must be zero:
   \[
   \sqrt{5} + 6 \sqrt{3} - 11 = 0
   \]

5. **Simplify the second equation**:
   Since \(\sqrt{5} + 6 \sqrt{3} - 11 = 0\) is a transcendental equation, it does not have simple rational or algebraic solutions. Therefore, the only practical solution in this case is:

   \[
   x = 0
   \]

### Final Answer:
The only root of the equation is \(x = 0\).

Would you like more details or have any questions?

### Related Questions:
1. How do we solve equations involving multiple square roots?
2. Can this equation have any irrational roots?
3. How do we solve transcendental equations?
4. What are the methods for factoring more complex expressions?
5. How do roots of square root terms affect the overall solution of an equation?

### Tip:
Always try to simplify square roots and combine like terms to make the equation easier to solve.

Explore the detailed step-by-step solution for the equation x√5 - 5x√4 + 6x√3 - x = 0. Learn how to simplify square root terms and solve transcendental equations using factoring techniques. Ideal for high school students (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