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(2x-3)(2x+3)/8=(x-4)^2/6+(x-2)^2/3

3 min read Jun 16, 2024
(2x-3)(2x+3)/8=(x-4)^2/6+(x-2)^2/3

Solving the Equation: (2x-3)(2x+3)/8=(x-4)^2/6+(x-2)^2/3

This article will guide you through the process of solving the equation: (2x-3)(2x+3)/8=(x-4)^2/6+(x-2)^2/3.

Simplifying the Equation

  1. Expand the squares:

    • (x-4)^2 = x^2 - 8x + 16
    • (x-2)^2 = x^2 - 4x + 4

  2. Expand the product:

    • (2x-3)(2x+3) = 4x^2 - 9

  3. Rewrite the equation with simplified terms: (4x^2 - 9)/8 = (x^2 - 8x + 16)/6 + (x^2 - 4x + 4)/3

Finding a Common Denominator

  1. Find the least common multiple (LCM) of the denominators 8, 6, and 3: The LCM is 24.

  2. Multiply each term by a fraction that will result in a denominator of 24:

    • (4x^2 - 9)/8 * (3/3) = (12x^2 - 27)/24
    • (x^2 - 8x + 16)/6 * (4/4) = (4x^2 - 32x + 64)/24
    • (x^2 - 4x + 4)/3 * (8/8) = (8x^2 - 32x + 32)/24

  3. Rewrite the equation with the common denominator: (12x^2 - 27)/24 = (4x^2 - 32x + 64)/24 + (8x^2 - 32x + 32)/24

Solving the Equation

  1. Multiply both sides of the equation by 24 to eliminate the denominators: 12x^2 - 27 = 4x^2 - 32x + 64 + 8x^2 - 32x + 32

  2. Combine like terms: 12x^2 - 27 = 12x^2 - 64x + 96

  3. Subtract 12x^2 from both sides: -27 = -64x + 96

  4. Subtract 96 from both sides: -123 = -64x

  5. Divide both sides by -64: x = 123/64

Solution

Therefore, the solution to the equation (2x-3)(2x+3)/8=(x-4)^2/6+(x-2)^2/3 is x = 123/64.

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