Here's a challenging equation that equals 5:

\[ \frac{2x^3 - 5x^2 + 3x + 7}{\sqrt{x + 3}} = 5 \]

This equation involves polynomial terms, fractions, and square roots.

### Steps to Solve

1. **Multiply both sides by \(\sqrt{x + 3}\) to eliminate the denominator:**
   \[
   2x^3 - 5x^2 + 3x + 7 = 5\sqrt{x + 3}
   \]

2. **Square both sides to eliminate the square root:**
   \[
   (2x^3 - 5x^2 + 3x + 7)^2 = 25(x + 3)
   \]

3. **Expand both sides:**
   \[
   (2x^3 - 5x^2 + 3x + 7)^2 = 25x + 75
   \]

4. **Expand the left side:**
   \[
   4x^6 - 20x^5 + 43x^4 - 70x^3 + 42x^2 + 49 = 25x + 75
   \]

5. **Bring all terms to one side to form a polynomial equation:**
   \[
   4x^6 - 20x^5 + 43x^4 - 70x^3 + 42x^2 - 25x - 26 = 0
   \]

6. **Solve the polynomial equation using numerical methods or factoring techniques if possible.**

This polynomial equation is quite complex and may require the use of numerical methods or computer software to solve for \( x \).

Would you like to delve into more details or have any questions about this example?

---

Here are some relative questions to expand your understanding:

1. What numerical methods can be used to solve high-degree polynomial equations?
2. How do you factorize complex polynomial equations?
3. What are the applications of polynomial equations in real-life scenarios?
4. How can graphing help in understanding the solutions of polynomial equations?
5. What is the importance of squaring both sides of an equation, and what are the potential pitfalls?
6. How do you handle equations involving both polynomial and radical terms?
7. What software tools are available to solve complex algebraic equations?
8. How does the Fundamental Theorem of Algebra relate to solving polynomial equations?

**Tip:** When dealing with high-degree polynomial equations, using graphing tools or numerical solvers can help visualize and approximate the solutions effectively.

Explore a challenging equation \( \frac{2x^3 - 5x^2 + 3x + 7}{\sqrt{x + 3}} = 5 \) with detailed steps and solutions. This problem involves polynomials, fractions, and square roots, suitable for advanced high school students and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation