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(2x+3)(2x+5)(x-1)(x-2)=30

4 min read Jun 16, 2024
(2x+3)(2x+5)(x-1)(x-2)=30

Solving the Equation: (2x+3)(2x+5)(x-1)(x-2) = 30

This equation involves a product of four factors set equal to a constant. To solve it, we'll follow these steps:

1. Expanding the Equation

First, expand the product of the factors:

  • Step 1: Expand the first two factors: (2x+3)(2x+5) = 4x² + 16x + 15

  • Step 2: Expand the last two factors: (x-1)(x-2) = x² - 3x + 2

  • Step 3: Combine the expansions: (4x² + 16x + 15)(x² - 3x + 2) = 30

  • Step 4: Expand the entire product: 4x⁴ - 12x³ + 8x² + 16x³ - 48x² + 32x + 15x² - 45x + 30 = 30

  • Step 5: Simplify by combining like terms: 4x⁴ + 4x³ - 25x² - 13x = 0

2. Finding the Solutions

Now we have a polynomial equation. To find its solutions, we can use factoring or the quadratic formula:

  • Factoring: The equation can be factored as: x(4x³ + 4x² - 25x - 13) = 0

  • Solution 1: One solution is immediately apparent: x = 0

  • Finding other solutions: The remaining solutions must satisfy the cubic equation: 4x³ + 4x² - 25x - 13 = 0

This cubic equation doesn't easily factor. We can use numerical methods like the Rational Root Theorem or graphing to find approximate solutions.

Alternatively: We can use the Rational Root Theorem to test potential rational roots. This theorem states that any rational root of a polynomial with integer coefficients must be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

In this case, the constant term is -13 and the leading coefficient is 4. The factors of -13 are ±1 and ±13, and the factors of 4 are ±1, ±2, and ±4. Therefore, the possible rational roots are ±1, ±13, ±1/2, ±13/2, ±1/4, and ±13/4.

By testing these values, we can find the approximate solutions to the cubic equation.

3. Conclusion

The equation (2x+3)(2x+5)(x-1)(x-2) = 30 has four solutions:

  • x = 0
  • The other three solutions are found by solving the cubic equation 4x³ + 4x² - 25x - 13 = 0 using numerical methods or the Rational Root Theorem.

Note: The exact solutions to the cubic equation may be irrational numbers.

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