(x+3)3-x(3x+1)2+(2x+1)(4x2-2x+1)=28

Jasonbradley

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Jun 16, 2024

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Solving the Algebraic Equation: (x+3)³ - x(3x+1)² + (2x+1)(4x² - 2x + 1) = 28

This article will guide you through the steps to solve the algebraic equation: (x+3)³ - x(3x+1)² + (2x+1)(4x² - 2x + 1) = 28.

Expanding the Equation

The first step is to expand the equation by multiplying out the terms:

1. (x+3)³: This can be expanded using the binomial theorem or by multiplying (x+3) by itself three times.
   (x+3)³ = (x+3)(x+3)(x+3) = x³ + 9x² + 27x + 27
2. x(3x+1)²: This can be expanded by multiplying (3x+1) by itself and then multiplying the result by x. x(3x+1)² = x(9x² + 6x + 1) = 9x³ + 6x² + x
3. (2x+1)(4x² - 2x + 1): This is a product of a sum and difference of squares. (2x+1)(4x² - 2x + 1) = (2x)³ + 1³ = 8x³ + 1

Now, the equation becomes:

x³ + 9x² + 27x + 27 - (9x³ + 6x² + x) + 8x³ + 1 = 28

Simplifying the Equation

Next, simplify the equation by combining like terms:

(x³ - 9x³ + 8x³) + (9x² - 6x²) + (27x - x) + (27 + 1 - 28) = 0

This simplifies to:

0x³ + 3x² + 26x = 0

Solving for x

Finally, we have a simplified quadratic equation: 3x² + 26x = 0. To solve for x, we can factor out a common factor of x:

x(3x + 26) = 0

This gives us two possible solutions:

• x = 0
• 3x + 26 = 0 => x = -26/3

Therefore, the solutions to the equation (x+3)³ - x(3x+1)² + (2x+1)(4x² - 2x + 1) = 28 are x = 0 and x = -26/3.