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(2x+3)2+(2x-3)2=(8x+6)(x-1)+22

2 min read Jun 16, 2024
(2x+3)2+(2x-3)2=(8x+6)(x-1)+22

Solving the Equation: (2x+3)2+(2x-3)2=(8x+6)(x-1)+22

This article will guide you through the process of solving the equation (2x+3)2+(2x-3)2=(8x+6)(x-1)+22. We will use algebraic manipulation to simplify the equation and isolate the variable x.

Simplifying the Equation

  1. Expand the Squares:

    • (2x+3)2 = (2x+3)(2x+3) = 4x² + 12x + 9
    • (2x-3)2 = (2x-3)(2x-3) = 4x² - 12x + 9

  2. Expand the Product:

    • (8x+6)(x-1) = 8x² - 8x + 6x - 6 = 8x² - 2x - 6

  3. Substitute the Expanded Terms:

    • Our equation now becomes: 4x² + 12x + 9 + 4x² - 12x + 9 = 8x² - 2x - 6 + 22

Solving for x

  1. Combine Like Terms:

    • 8x² + 18 = 8x² - 2x + 16

  2. Isolate x:

    • Subtract 8x² from both sides: 18 = -2x + 16
    • Subtract 16 from both sides: 2 = -2x

  3. Solve for x:

    • Divide both sides by -2: x = -1

Conclusion

Therefore, the solution to the equation (2x+3)2+(2x-3)2=(8x+6)(x-1)+22 is x = -1. You can always verify your answer by substituting this value back into the original equation.

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