(x-3)(x^2+3x+9)+x(x+2)(2-x)=1

Jasonbradley

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Jun 17, 2024

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Solving the Equation: (x-3)(x^2+3x+9)+x(x+2)(2-x)=1

This article will guide you through the process of solving the equation: (x-3)(x^2+3x+9)+x(x+2)(2-x)=1

Understanding the Equation

The equation consists of two parts:

1. (x-3)(x^2+3x+9): This is the factored form of the difference of cubes. It can be expanded to x³ - 27.
2. x(x+2)(2-x): This can be simplified by first multiplying (x+2)(2-x) which gives -x² + 4.
   • Then, multiply this result by x to get -x³ + 4x.

Solving the Equation

1. Expand and Simplify:
   • Substitute the expanded forms back into the original equation:
     • (x³ - 27) + (-x³ + 4x) = 1
   • Simplify the equation:
     • 4x - 27 = 1
2. Isolate the variable 'x':
   • Add 27 to both sides:
     • 4x = 28
3. Solve for 'x':
   • Divide both sides by 4:
     • x = 7

Solution

Therefore, the solution to the equation (x-3)(x^2+3x+9)+x(x+2)(2-x)=1 is x = 7.

Verification

To verify the solution, substitute x = 7 back into the original equation:

(7 - 3)(7² + 3(7) + 9) + 7(7 + 2)(2 - 7) = 1

Simplifying the expression:

(4)(49 + 21 + 9) + 7(9)(-5) = 1

79(4) - 315 = 1

316 - 315 = 1

1 = 1

This confirms that x = 7 is indeed the correct solution to the equation.