Based on a zero product property which of the following statements must be true about the quadratic equation 2x - 3 * x + 2 = 0

Question

Neither of the factors two x minus 3 or x + 2 must equal 0 for the entire function of of function to equal zero at least one of the factors must equal 0 no value of x will make the function equal 0 or both factors must equal 0 for the entire function to

Answer 1

To analyze the quadratic equation \( (2x - 3)(x + 2) = 0 \) using the zero product property, we can break down the statement:

1. Zero Product Property: This property states that if the product of two factors equals zero, then at least one of the factors must equal zero. In other words, for the equation \( A \cdot B = 0 \), it follows that either \( A = 0 \) or \( B = 0 \) (or both).

Given the factors of the equation \( (2x - 3)(x + 2) = 0 \):

• For the product to equal zero, at least one of the following must be true:
  • \( 2x - 3 = 0 \)
  • \( x + 2 = 0 \)

Thus, to answer the question, the statement that must be true is:

"At least one of the factors must equal 0."

The other statements are not true:

• "Neither of the factors must equal 0" is incorrect because we know at least one must.
• "No value of x will make the function equal 0" is incorrect because solutions exist where the function equals 0.
• "Both factors must equal 0" is incorrect; only one factor needs to equal 0 for the product to be zero.

So, the correct and true statement is: "At least one of the factors must equal 0."