(x-3)(x^2+6x+9)=7(x+3)

Jasonbradley

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Jun 17, 2024

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Solving the Equation: (x-3)(x^2+6x+9) = 7(x+3)

This equation presents a polynomial equation that we can solve using algebraic manipulation. Here's a step-by-step solution:

1. Expand both sides of the equation:

• On the left side, we recognize that (x^2 + 6x + 9) is a perfect square trinomial: (x+3)^2.
• Therefore, the left side becomes: (x-3)(x+3)^2
• The right side remains: 7(x+3)

2. Simplify the equation:

• We now have: (x-3)(x+3)^2 = 7(x+3)
• Notice that (x+3) is a common factor on both sides. Let's factor it out:
  • (x+3) [(x-3)(x+3) - 7] = 0

3. Solve for the roots:

• The equation is now in factored form. To find the roots (solutions), we set each factor to zero:
  • (x+3) = 0 --> x = -3
  • (x-3)(x+3) - 7 = 0 --> x^2 - 9 - 7 = 0 --> x^2 - 16 = 0 --> (x-4)(x+4) = 0
  • From the second factor, we get: x = 4 and x = -4

4. Solutions:

• Therefore, the solutions to the equation (x-3)(x^2+6x+9) = 7(x+3) are:
  • x = -3
  • x = 4
  • x = -4

Important Note: It is always crucial to check the solutions obtained by plugging them back into the original equation to ensure they are valid. In this case, all three solutions satisfy the original equation.