(x+2)(3x+4)(3x+7)(x+3)=2400

3 min read Jun 16, 2024

Solving the Equation (x+2)(3x+4)(3x+7)(x+3)=2400

This equation presents a challenge in its current form. Let's break down how to solve it:

Understanding the Problem

We have a quartic equation (an equation with the highest power of x being 4). Solving quartic equations can be complex and often requires specific methods.

Simplifying the Equation

Expand the equation: Multiply out the factors on the left side. This will result in a polynomial equation of the form:
```
ax^4 + bx^3 + cx^2 + dx + e = 2400
```
Move the constant term: Subtract 2400 from both sides to get a standard form:
```
ax^4 + bx^3 + cx^2 + dx - 2400 = 0
```

Methods for Solving

Here are a few approaches to solving the equation:

Factoring:
- Try to factor the polynomial on the left side. This is often the simplest approach, but it may not always be possible.
- Look for common factors or patterns that can be exploited.
Rational Root Theorem:
- This theorem helps identify potential rational roots (roots that can be expressed as fractions).
- Once you find a root, you can divide the polynomial by (x - root) to reduce the degree of the equation.
Numerical Methods:
- If factoring and the Rational Root Theorem fail, you can use numerical methods like the Newton-Raphson method or graphing calculators to approximate the roots.

Finding the Solutions

While the specific solution steps will depend on the chosen method, the goal is to find the values of x that satisfy the equation.

Important Notes:

Quartic equations can have up to four distinct solutions (roots).
Some solutions might be real numbers, while others might be complex numbers.
It's possible that the equation might have no real solutions.

Conclusion

Solving the equation (x+2)(3x+4)(3x+7)(x+3)=2400 involves simplifying the equation, applying appropriate methods like factoring, the Rational Root Theorem, or numerical methods, and ultimately finding the values of x that make the equation true.