(2x-1)(3x+2)=0

Jasonbradley

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Jun 16, 2024

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Solving the Equation (2x-1)(3x+2) = 0

This equation represents a quadratic equation in factored form. To solve for the values of x that satisfy the equation, we can use 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.

Applying the Zero Product Property

1. Set each factor equal to zero:

   • 2x - 1 = 0
   • 3x + 2 = 0

2. Solve each equation for x:

   • 2x = 1

   • x = 1/2

   • 3x = -2

   • x = -2/3

Solutions

Therefore, the solutions to the equation (2x-1)(3x+2) = 0 are:

• x = 1/2
• x = -2/3

These values represent the x-intercepts of the parabola that the equation represents.

Verifying the Solutions

To verify our solutions, we can substitute each value back into the original equation:

• For x = 1/2:

  • (2(1/2) - 1)(3(1/2) + 2) = (1 - 1)(3/2 + 2) = 0 * 7/2 = 0
  • This confirms that x = 1/2 is a valid solution.

• For x = -2/3:

  • (2(-2/3) - 1)(3(-2/3) + 2) = (-4/3 - 1)(-2 + 2) = (-7/3) * 0 = 0
  • This confirms that x = -2/3 is a valid solution.

In conclusion, the equation (2x-1)(3x+2) = 0 has two solutions: x = 1/2 and x = -2/3.