(x+7)(x-7)-(3x-1)(x+1)=4-2x^2

Jasonbradley

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Jun 17, 2024

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Solving the Equation: (x+7)(x-7)-(3x-1)(x+1)=4-2x^2

This article will guide you through solving the equation (x+7)(x-7)-(3x-1)(x+1)=4-2x^2. We will simplify the equation, solve for 'x', and verify our solution.

Simplifying the Equation

1. Expand the products:

   • (x+7)(x-7) = x² - 49 (using the difference of squares pattern)
   • (3x-1)(x+1) = 3x² + 2x - 1

2. Substitute the expanded products into the original equation:

   • x² - 49 - (3x² + 2x - 1) = 4 - 2x²

3. Distribute the negative sign:

   • x² - 49 - 3x² - 2x + 1 = 4 - 2x²

4. Combine like terms:

   • -2x² - 2x - 48 = 4 - 2x²

5. Simplify further:

   • -2x - 48 = 4

Solving for 'x'

1. Isolate the 'x' term:

   • -2x = 52

2. Solve for 'x':

   • x = -26

Verifying the Solution

1. Substitute x = -26 back into the original equation:

   • (-26 + 7)(-26 - 7) - (3(-26) - 1)(-26 + 1) = 4 - 2(-26)²

2. Simplify both sides of the equation:

   • (-19)(-33) - (-79)(-25) = 4 - 2(676)
   • 627 - 1975 = 4 - 1352
   • -1348 = -1348

Since both sides of the equation are equal, we have verified that x = -26 is the correct solution.

Therefore, the solution to the equation (x+7)(x-7)-(3x-1)(x+1)=4-2x² is x = -26.