Use the Zero Product Property to find the value of x that makes the following statement true: 2⋅(x−3)=0

Question

Use the Zero Product Property to find the value of x that makes the following statement true: 2⋅(x−3)=0
.(1 point)
The value that makes the statement true is x= (    )

GPT-4o mini · Answer

To solve the equation $ 2 \cdot (x - 3) = 0 $ using the Zero Product Property, we first set the equation equal to zero. Since $ 2 $ is a non-zero constant, we can find the solution by focusing on the factor $ (x - 3) $. According to the Zero Product Property, if the product of two factors is zero, at least one of the factors must be zero. Therefore, we can set the factor equal to zero: $$ x - 3 = 0 $$ Now, we can solve for $ x $: $$ x = 3 $$ Thus, the value that makes the statement true is $ x = 3 $. So, the final answer is: The value that makes the statement true is $ x = \mathbf{3} $.

x · Answer

Option 1 (x−2)(x+3)=7

Option 2 x(x+3)=0

Option 3 (x+3)^2−1=0

It is only possible to use the Zero Product Property on one of the options in this table. Which option can the Zero Product Property be used on to find the zeros of the function?

(1 point)
Option

GPT-4o mini · Answer

The Zero Product Property states that if the product of multiple factors equals zero, then at least one of those factors must also equal zero. Let's examine each of the given options:

Option 1:

$(x - 2)(x + 3) = 7$

This equation does not equal zero, so the Zero Product Property cannot be directly applied here.

Option 2:

$x(x + 3) = 0$

This equation is already set to zero, meaning we can use the Zero Product Property. By setting each factor equal to zero:

$x = 0$
$x + 3 = 0 \rightarrow x = -3$

Both values, $x = 0$ and $x = -3$, are the solutions.

Option 3:

$(x + 3)^2 - 1 = 0$

This equation can be rearranged, but it is not in a product form equal to zero. We would need to factor or manipulate it further to find the zeros, which does not directly apply the Zero Product Property.

Conclusion:

Option 2 $ x(x + 3) = 0 $ is the only option where the Zero Product Property can be used to find the zeros of the function.

Therefore, the answer is: Option 2.

x · Answer

Solve the factored quadratic equation (x−5)(x+3)=0 .(1 point) The smaller solution is x= ( ) , and the larger solution is x= ( )

GPT-4o mini · Answer

To solve the factored quadratic equation $(x - 5)(x + 3) = 0$, we can use the Zero Product Property. This property states that if the product of two factors equals zero, then at least one of the factors must be equal to zero.

We set each factor equal to zero:

$x - 5 = 0$
$x + 3 = 0$

Now, solve each equation:

For $x - 5 = 0$: \[ x = 5 \]
For $x + 3 = 0$: \[ x = -3 \]

Now we have the two solutions: $x = 5$ and $x = -3$.

To summarize:

The smaller solution is $x = -3$.
The larger solution is $x = 5$.

Therefore, the answers are: The smaller solution is $x = (-3)$, and the larger solution is $x = (5)$.

x · Answer

To begin a football game, a kicker has to kick off. The football follows the path h(x)=−1/30x (x−61)
, where h(x)
is the height of the football and x is the horizontal distance it has traveled in yards. Solve the equation to determine how far the ball will have traveled when it hits the ground.

0=−13/0x (x−61)

(1 point)
The ball will have traveled ( ) yards before hitting the ground.

GPT-4o mini · Answer

To determine how far the football travels before it hits the ground, we need to find the values of $ x $ when the height $ h(x) $ is equal to zero. The height of the football is given by the equation:

\[ h(x) = -\frac{1}{30}x(x - 61) \]

We need to set this equation equal to zero to find when the ball hits the ground:

\[ 0 = -\frac{1}{30}x(x - 61) \]

To eliminate the fraction, we can multiply both sides of the equation by $-30$:

\[ 0 = x(x - 61) \]

Now, apply the Zero Product Property by setting each factor equal to zero:

$ x = 0 $
$ x - 61 = 0 \Rightarrow x = 61 $

The football will be at ground level (height $ h(x) = 0 $) when $ x = 0 $ (at the moment of the kick-off) and when $ x = 61 $ yards.

Thus, the ball will have traveled $ x = 61 $ yards before hitting the ground.

The final answer is: The ball will have traveled $ \mathbf{61} $ yards before hitting the ground.

x · Answer

(x+3)(x+4)=0
x−3=0 x+4=0
x=3 x=−4While 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?(1 point)
Responses

No, x+4=0 in the second row should be x−4=0.
No, x plus 4 equals 0 in the second row should be x minus 4 equals 0 .

No, x−3=0 in the second row should be x+3=0.
No, x minus 3 equals 0 in the second row should be x plus 3 equals 0 .

No, he should have a third column where he should set x=0.
No, he should have a third column where he should set x equals 0 .

Yes, the work is accurate.

GPT-4o mini · Answer

Let's review Oliver's work step by step for the equation $(x + 3)(x + 4) = 0$.

Setting Each Factor to Zero:

From $(x + 3) = 0$, we get: \[ x = -3 \]
From $(x + 4) = 0$, we get: \[ x = -4 \]

Oliver's Steps:

It seems Oliver wrote:
- $x - 3 = 0$ for the first factor, which is incorrect because it should have been $x + 3 = 0$.
- He correctly set $x + 4 = 0$, but later, he interpreted that as $x = -4$, but stated the wrong equation in between.

Conclusion:

Oliver's work has inaccuracies regarding the first factor, where he wrote $x - 3 = 0$ instead of $x + 3 = 0$.

So, the correct answer from the provided options regarding Oliver's accuracy is: No, $x - 3 = 0$ in the second row should be $x + 3 = 0$.

x · Answer

Based on the Zero Product Property, which of the following statements must be true about the quadratic equation (2x−3)(x+2)=0 ?(1 point)
Responses

Neither of the factors, (2x−3) or (x+2), must equal 0 for the entire function to equal 0.
Neither 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) or (x+2), must equal 0 for the entire function to equal 0.
Both 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.

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.

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

GPT-4o mini · Answer

To solve the quadratic equation $ (2x - 3)(x + 2) = 0 $ using the Zero Product Property, we know that if the product of two factors is equal to zero, then at least one of the factors must also equal zero. This means we can conclude that: - **At least one of the factors, $ (2x - 3) $ or $ (x + 2) $, must equal 0 for the entire function to equal 0.** Thus, the correct statement based on the Zero Product Property is: **At least one of the factors, $ (2x - 3) $ or $ (x + 2) $, must equal 0 for the entire function to equal 0.**