Based on the Zero Product Property, which of the following statements must be true about the quadratic equation (2x−3)(x+2)=0

Question

?(1 point)
Responses

At least one of the factors, (2x−3)
or ​(x+2)
, must equal 0 for the entire function to equal 0.
At least one of the factors, left parenthesis 2 x minus 3 right parenthesis or left parenthesis x plus 2 right parenthesis , must equal 0 for the entire function to equal 0.

Both factors, (2x−3)
and ​(x+2)
, must equal 0 for the entire function to equal 0.
Both factors, left parenthesis 2 x minus 3 right parenthesis and left parenthesis x plus 2 right parenthesis , must equal 0 for the entire function to equal 0.

Neither of the factors, (2x−3)
nor ​(x+2)
, must equal 0 for the entire function to equal 0.
Neither of the factors, left parenthesis 2 x minus 3 right parenthesis nor left parenthesis x plus 2 right parenthesis , must equal 0 for the entire function to equal 0.

No values of x will make the function equal 0.
No values of x will make the function equal 0.

Answer 1

Based on the Zero Product Property, the correct statement about the quadratic equation \((2x−3)(x+2)=0\) is:

At least one of the factors, (2x−3) or (x+2), must equal 0 for the entire function to equal 0.

This means that in order for the product of those two factors to equal zero, at least one of the factors must itself be equal to zero. Therefore, the corresponding values of \(x\) can be found by solving each factor separately:

1. \(2x - 3 = 0\) leads to \(x = \frac{3}{2}\)
2. \(x + 2 = 0\) leads to \(x = -2\)

Thus, the equation has two solutions, but only one of the factors needs to be zero for the entire function to equal zero.