(x-1)(2x+7)(x^2+2)=0

Jasonbradley

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Jun 17, 2024

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Solving the Equation: (x-1)(2x+7)(x^2+2) = 0

This equation is a polynomial equation in factored form, making it relatively simple to solve.

Understanding the Principle:

The key principle behind solving this equation is the Zero Product Property. This property states that if the product of two or more factors is zero, then at least one of the factors must be zero.

Solving the Equation:

1. Set each factor equal to zero:

   • x - 1 = 0
   • 2x + 7 = 0
   • x² + 2 = 0

2. Solve for x in each equation:

   • x - 1 = 0
     • Add 1 to both sides: x = 1
   • 2x + 7 = 0
     • Subtract 7 from both sides: 2x = -7
     • Divide both sides by 2: x = -7/2
   • x² + 2 = 0
     • Subtract 2 from both sides: x² = -2
     • Since the square of any real number cannot be negative, this equation has no real solutions.

Therefore, the solutions to the equation (x-1)(2x+7)(x^2+2) = 0 are:

• x = 1
• x = -7/2

In conclusion, the equation has two real solutions: 1 and -7/2. It's important to note that while the equation has no real solutions for the factor (x² + 2), it does have complex solutions.