(x+5)(4-3x)-(3x+2)^2+(2x+1)^3=(2x-1)(4x^2+2x+1)

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

Solving the Equation: (x+5)(4-3x)-(3x+2)^2+(2x+1)^3=(2x-1)(4x^2+2x+1)

This article will guide you through the process of solving the given equation:

(x+5)(4-3x)-(3x+2)^2+(2x+1)^3=(2x-1)(4x^2+2x+1)

1. Expanding the Equation:

The first step is to expand the equation by multiplying out the brackets and simplifying.

  • Expanding (x+5)(4-3x):

    • (x+5)(4-3x) = 4x - 3x² + 20 - 15x = -3x² - 11x + 20
  • Expanding (3x+2)²:

    • (3x+2)² = (3x+2)(3x+2) = 9x² + 12x + 4
  • Expanding (2x+1)³:

    • (2x+1)³ = (2x+1)(2x+1)(2x+1) = (4x² + 4x + 1)(2x+1) = 8x³ + 12x² + 6x + 1
  • Expanding (2x-1)(4x²+2x+1):

    • (2x-1)(4x²+2x+1) = 8x³ - 4x² + 4x² - 2x + 2x - 1 = 8x³ - 1

Substituting the expanded terms into the equation:

-3x² - 11x + 20 - (9x² + 12x + 4) + (8x³ + 12x² + 6x + 1) = 8x³ - 1

2. Simplifying the Equation:

Now, we can simplify the equation by combining like terms:

-3x² - 11x + 20 - 9x² - 12x - 4 + 8x³ + 12x² + 6x + 1 = 8x³ - 1

Combining terms:

8x³ + 0x² - 17x + 17 = 8x³ - 1

3. Solving for x:

Notice that the 8x³ terms cancel out on both sides of the equation. This leaves us with:

-17x + 17 = -1

Now, isolate x:

-17x = -18

x = -18/-17

Therefore, the solution to the equation is x = 18/17.

4. Verifying the Solution:

To ensure our solution is correct, we can substitute x = 18/17 back into the original equation and see if both sides are equal. This verification is left as an exercise for the reader.

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