The Zero Product Property states that if the product of two factors is equal to zero, then at least one of the factors must be equal to zero. For the quadratic equation \((2x - 3)(x + 2) = 0\), we can apply this property as follows:
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Set each factor equal to zero:
- \(2x - 3 = 0\)
- \(x + 2 = 0\)
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Solve each equation:
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For \(2x - 3 = 0\): \[ 2x = 3 \ x = \frac{3}{2} \]
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For \(x + 2 = 0\): \[ x = -2 \]
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Thus, the solutions (or roots) of the quadratic equation are \(x = \frac{3}{2}\) and \(x = -2\).
Therefore, the statements based on the Zero Product Property that must be true about the quadratic equation \((2x - 3)(x + 2) = 0\) are:
- \(2x - 3 = 0\) or \(x + 2 = 0\)
- The solutions to the equation are \(x = \frac{3}{2}\) and \(x = -2\).
These are the conclusions that can be drawn from applying the Zero Product Property to this equation.