Question

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

Answer 1

[AI-generated answer — Bot GPT 3.5]

Based on the Zero Product Property, the statement "if a product of two factors is equal to zero, then at least one of the factors must be equal to zero" applies.

In the case of the quadratic equation (2x−3)(x+2)=0, to satisfy the Zero Product Property, either (2x - 3) must be equal to zero or (x + 2) must be equal to zero.

Therefore, the following statements must be true:

1. 2x - 3 = 0
or
2. x + 2 = 0

Answer 2

[AI-generated answer — Step-by-Step Bot]

The Zero Product Property states that if the product of two factors is zero, then at least one of the factors must be zero.

In the quadratic equation (2x−3)(x+2)=0, the product of the two factors is zero, so we can set each factor equal to zero and solve for x.

Setting 2x - 3 = 0:
2x = 3
x = 3/2

Setting x + 2 = 0:
x = -2

Therefore, the solutions to the quadratic equation (2x−3)(x+2)=0 are x = 3/2 and x = -2.

Answer 3

[AI-generated answer — Explain Bot]

To understand the Zero Product Property and determine which statement must be true about the given quadratic equation, let's break it down step by step.

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 zero.

So, looking at the given quadratic equation:

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

To apply the Zero Product Property, we set each factor equal to zero and solve for x.

Setting the first factor equal to zero:

2x - 3 = 0

Adding 3 to both sides:

2x = 3

Dividing both sides by 2:

x = 3/2

Now, setting the second factor equal to zero:

x + 2 = 0

Subtracting 2 from both sides:

x = -2

So, we have two possible solutions for x: x = 3/2 and x = -2.

Now, let's analyze the given statements and see which one must be true:

Statement 1: x = 3/2
Statement 2: x = -2
Statement 3: x ≠ 3 and x ≠ -2

Based on our calculations, it is evident that both x = 3/2 and x = -2 are valid solutions to the quadratic equation. Therefore, statement 3 must be false, and either statement 1 or statement 2 must be the correct answer.

In conclusion, based on the Zero Product Property, the statement that must be true about the quadratic equation (2x - 3)(x + 2) = 0 is either statement 1: x = 3/2 or statement 2: x = -2.