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Use the table to answer the question.

(x+3)(x+4)=0
x−3=0 x+4=0
x=3 x=−4 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?(1 p

7 answers

Let's verify Oliver's work concerning the equation \((x+3)(x+4) = 0\) using the Zero Product Property.

According to the Zero Product Property, if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we have:

  1. Set each factor equal to zero: \[ x + 3 = 0 \quad \text{or} \quad x + 4 = 0 \]

  2. Solving each equation:

    • From \(x + 3 = 0\): \[ x = -3 \]
    • From \(x + 4 = 0\): \[ x = -4 \]

Thus, the solutions for \((x+3)(x+4) = 0\) are \(x = -3\) and \(x = -4\).

Now let's review Oliver's work:

  • He listed \(x - 3 = 0\) which is incorrect because it corresponds to a different equation.
  • He correctly found \(x + 4 = 0\) leading to \(x = -4\).
  • He stated \(x = 3\) which is also incorrect.

Oliver's solutions for \(x\) should be \(x = -3\) and \(x = -4\). Therefore, his work is not accurate.

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

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.

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

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.

At least one of the factors, (2x−3) or ​(x+2), must equal 0 for the entire function to equal 0.

Based on the Zero Product Property, the correct statement regarding the quadratic equation \((2x - 3)(x + 2) = 0\) is:

At least one of the factors, \((2x - 3)\) or \((x + 2)\), must equal 0 for the entire function to equal 0.

This statement reflects the principle that if the product of two factors is zero, then at least one of the factors must be zero.

The path of a soccer ball can be modeled by the equation h(x)=−12x(x−20), where h(x) is the height of the ball, and x is the horizontal distance the ball has traveled. To determine how far the ball has traveled when it hits the ground, Leon set the height equal to 0 and obtained the equation: 0=−12x(x−20). How can he use the Zero Product Property to find the horizontal distance the ball has traveled when the height of the soccer ball equals 0?(1 point)
Responses

Plug in 0 for all the x terms in the function.
Plug in 0 for all the x terms in the function.

He cannot use the Zero Product Property because the height will never be 0.
He cannot use the Zero Product Property because the height will never be 0.

Cancel out the x and then multiply −12 and −20.
Cancel out the x and then multiply negative Start Fraction 1 over 2 End Fraction and negative 20 .

Solve the two equations: −12x=0 and x−20=0.

To find the horizontal distance the ball has traveled when it hits the ground (when \(h(x) = 0\)), Leon can use the Zero Product Property on the equation he derived:

\[ 0 = -12x(x - 20). \]

Using the Zero Product Property, he can set each factor equal to zero:

  1. \(-12x = 0\)
  2. \(x - 20 = 0\)

Thus, to find the values of \(x\) that make the equation true, Leon should solve these two equations.

So the correct response is:

Solve the two equations: −12x=0 and x−20=0.

Solve the factored quadratic equation 0=−13x(x−6).(1 point)
Responses

The solution set is x={0,−6}.
The solution set is x equals left brace neagtive 0 comma negative 6 right brace .

The solution set is x={13,6}.
The solution set is x equals left brace Start Fraction 1 over 3 End Fraction comma 6 right brace .

The solution set is x={0,6}.
The solution set is x equals left brace 0 comma 6 right brace .

The solution set is x={−6,3}.

To solve the factored quadratic equation

\[ 0 = -13x(x - 6), \]

we can use the Zero Product Property, which states that if the product is zero, then at least one of the factors must be zero.

The factors are:

  1. \(-13x = 0\)
  2. \(x - 6 = 0\)

Now we will solve each equation:

  1. For \(-13x = 0\): \[ x = 0. \]

  2. For \(x - 6 = 0\): \[ x = 6. \]

Therefore, the solution set is:

\[ x = {0, 6}. \]

So, the correct response is:

The solution set is x={0,6}.