While using the Zero Product Property to find the values of x that make the quadratic equation (x+3)(x+4) equals 0, Oliver completed the work provided in the table. Is Oliver’s work accurate?
To determine if Oliver's work is accurate, let's examine each step.
Oliver started with the quadratic equation (x+3)(x+4) = 0.
Next, he applied the Zero Product Property, which states that if a product of two factors equals zero, then at least one of the factors must equal zero. So, he set each factor equal to zero:
(x + 3) = 0 and (x + 4) = 0.
Next, Oliver solved each equation individually:
x + 3 = 0 ---> x = -3
x + 4 = 0 ---> x = -4
Let's check these values:
If we substitute x = -3 into the quadratic equation, we get:
(-3 + 3)(-3 + 4) = 0
(0)(1) = 0
0 = 0
This checks out. Similarly, if we substitute x = -4:
(-4 + 3)(-4 + 4) = 0
(-1)(0) = 0
0 = 0
This also checks out.
Therefore, Oliver's work is accurate. The values x = -3 and x = -4 make the quadratic equation (x+3)(x+4) equal to 0.
Based on the Zero Product Property, which of the following statements must be true about the quadratic equation (2x−3)(x+2)=0 ?
According to the Zero Product Property, if a product of two factors equals zero, then at least one of the factors must equal zero.
In the given quadratic equation (2x−3)(x+2) = 0, we can apply the Zero Product Property to determine the possible values of x.
Setting the first factor equal to zero: 2x - 3 = 0
Solving for x:
2x = 3
x = 3/2
Setting the second factor equal to zero: x + 2 = 0
Solving for x:
x = -2
Therefore, based on the Zero Product Property, the equation (2x−3)(x+2) = 0 implies that either x = 3/2 or x = -2 must be true.
To determine if Oliver's work is accurate, we need to check if his steps align with the correct usage of the Zero Product Property.
The Zero Product Property states that if AB = 0, then either A = 0 or B = 0. In this case, we have (x+3)(x+4) = 0, which implies that either (x+3) = 0 or (x+4) = 0.
Let's evaluate Oliver's work:
| Equation | Work | Correct? |
|----------------|------------------|----------|
| (x+3)(x+4) = 0 | x+3 = 0 | No |
| | x = -3 | No |
| (x+3)(x+4) = 0 | x+4 = 0 | No |
| | x = -4 | No |
Based on Oliver's work, it seems that his steps are incorrect. He mistakenly isolated the first factor, x+3, and the second factor, x+4, separately and then solved for x. This approach is not valid when using the Zero Product Property.
Therefore, Oliver's work is not accurate.
To determine if Oliver's work is accurate, let's review the Zero Product Property and compare it with his calculations.
The Zero Product Property states that if the product of two or more factors is equal to zero, then at least one of the factors must be equal to zero.
In this case, the given quadratic equation is (x+3)(x+4) = 0. To find the values of x that make this equation true, we need to set each factor equal to zero and solve for x.
Let's go step by step through Oliver's work:
| Step | Oliver's Work |
|-----------------|------------------------------------|
| Step 1 | (x+3)(x+4) = 0 |
| Step 2 | x + 3 = 0 or x + 4 = 0 |
| Step 3 | x = -3 or x = -4 |
Based on the table, Oliver has correctly factored the quadratic equation using the Zero Product Property. He correctly set each factor equal to zero in Step 2, and then solved for x in Step 3.
Therefore, Oliver's work is accurate. The values of x that make the quadratic equation (x+3)(x+4) = 0 true are x = -3 and x = -4.