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

4 min read Jun 16, 2024
(vi) (x+2)(3x+4)(3x+7)(x+3)=2400

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

This problem involves solving a fourth-degree polynomial equation. Here's a step-by-step approach to find the solutions:

1. Expand the Equation

Begin by expanding the left side of the equation:

  • First, multiply the first two factors and the last two factors:
    • (x+2)(3x+4) = 3x² + 10x + 8
    • (3x+7)(x+3) = 3x² + 16x + 21
  • Then multiply the resulting expressions:
    • (3x² + 10x + 8)(3x² + 16x + 21) = 9x⁴ + 98x³ + 289x² + 272x + 168

Now the equation becomes: 9x⁴ + 98x³ + 289x² + 272x + 168 = 2400

2. Rearrange the Equation

Subtract 2400 from both sides to set the equation to zero:

9x⁴ + 98x³ + 289x² + 272x - 2232 = 0

3. Finding the Solutions

Unfortunately, there's no straightforward algebraic method to solve a fourth-degree polynomial equation. Here are common approaches:

  • Factoring: Try to factor the equation. This might be possible if there are rational roots. However, it's often difficult to factor higher-degree polynomials directly.
  • Rational Root Theorem: This theorem helps find potential rational roots. If a rational number p/q is a root, then p must be a factor of the constant term (-2232) and q must be a factor of the leading coefficient (9).
  • Numerical Methods: Use numerical methods like the Newton-Raphson method or graphing calculators to approximate the solutions.

4. Using Numerical Methods (Example with Newton-Raphson)

The Newton-Raphson method is an iterative process. To use it, you need:

  • The function: f(x) = 9x⁴ + 98x³ + 289x² + 272x - 2232
  • Its derivative: f'(x) = 36x³ + 294x² + 578x + 272

Steps:

  1. Choose an initial guess (x₀). You might try a value close to where the graph of the function crosses the x-axis.
  2. Iterate using the formula: xₙ₊₁ = xₙ - f(xₙ) / f'(xₙ)
  3. Repeat step 2 until the value of x converges to a solution.

Note: The Newton-Raphson method may not always converge, and you might need to try different initial guesses.

Conclusion

Solving the equation (vi) (x+2)(3x+4)(3x+7)(x+3)=2400 involves expanding the equation, rearranging it, and then using numerical methods or potentially factoring to find the solutions. While factoring can be challenging for higher-degree polynomials, numerical methods like the Newton-Raphson method offer a robust way to approximate the solutions.

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