(x^2+2x-8)(x^2-8x+15)=(x^2-x-20)(x^2+5x-14)

Jasonbradley

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Jun 17, 2024

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Solving the Equation: (x^2+2x-8)(x^2-8x+15)=(x^2-x-20)(x^2+5x-14)

This equation involves a product of two quadratic expressions on each side. To solve it effectively, we can follow these steps:

1. Expand Both Sides of the Equation

First, expand the products on both sides of the equation using the distributive property or the FOIL method:

• Left-hand side:
  • (x^2+2x-8)(x^2-8x+15) = x^4 - 6x^3 - 29x^2 + 126x - 120
• Right-hand side:
  • (x^2-x-20)(x^2+5x-14) = x^4 + 4x^3 - 49x^2 - 60x + 280

2. Simplify the Equation

Now, we have a simplified equation:

x^4 - 6x^3 - 29x^2 + 126x - 120 = x^4 + 4x^3 - 49x^2 - 60x + 280

3. Combine Like Terms

Bring all the terms to one side of the equation:

-10x^3 + 20x^2 + 186x - 400 = 0

4. Factor the Equation

Now we need to factor the equation. This might seem tricky, but notice that all the coefficients are divisible by 2:

-5x^3 + 10x^2 + 93x - 200 = 0

We can factor this by grouping. The process involves finding a common factor among the first two terms and the last two terms.

-5x^3 + 10x^2 + 93x - 200 = 0 -5x^2(x - 2) + 93(x - 2) = 0 (-5x^2 + 93)(x - 2) = 0

Now, the equation is in factored form.

5. Solve for x

For the product of two factors to equal zero, at least one of them must be zero. This gives us two possible solutions:

• -5x^2 + 93 = 0
  • 5x^2 = 93
  • x^2 = 93/5
  • x = ±√(93/5)
• x - 2 = 0
  • x = 2

6. Solutions

Therefore, the solutions to the equation (x^2+2x-8)(x^2-8x+15)=(x^2-x-20)(x^2+5x-14) are:

• x = 2
• x = √(93/5)
• x = -√(93/5)

Note: It's always a good idea to check your solutions by substituting them back into the original equation to ensure they are valid.