Based on the Zero Product Property, which of the following statements must be true about the quadratic equation (2x−3)(x+2)=0
Based on the Zero Product Property, for a product to equal zero, at least one of the factors must equal zero. In the given quadratic equation (2x−3)(x+2)=0, for the product to equal zero, either (2x-3) must equal zero or (x+2) must equal zero.
Therefore, the following statements must be true:
- 2x - 3 = 0
- x + 2 = 0
To determine which statements are true about the quadratic equation (2x - 3)(x + 2) = 0 using the Zero Product Property, you need to set each factor equal to zero and solve for x.
Step 1: Set the first factor equal to zero:
2x - 3 = 0
Add 3 to both sides:
2x = 3
Divide both sides by 2:
x = 3/2
Step 2: Set the second factor equal to zero:
x + 2 = 0
Subtract 2 from both sides:
x = -2
Now that you have the solutions for x, you can determine the statements that are true:
Statement 1: x = 3/2
Since you found x = 3/2 as one of the solutions, the statement is true.
Statement 2: x = -2
Since you found x = -2 as one of the solutions, the statement is true.
Statement 3: x = 0
You did not find x = 0 as one of the solutions, so the statement is not true.
Therefore, the true statements about the quadratic equation (2x - 3)(x + 2) = 0 using the Zero Product Property are "x = 3/2" and "x = -2".
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.
In the given equation, (2x − 3)(x + 2) = 0, the product of the two factors is equal to zero. Therefore, we can 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 or x = 1.5
Setting the second factor equal to zero:
x + 2 = 0
Subtracting 2 from both sides:
x = -2
Therefore, the solutions for the quadratic equation (2x − 3)(x + 2) = 0 are x = 3/2, x = 1.5, and x = -2.