Find the values of x that make the following equation equal zero: (x+4)(x−2)=0.(1 point)

Question

Responses

x=−4 and x=2
x equals negative 4 and x equals 2

x=−2 and x=4
x equals negative 2 and x equals 4

x=−2 and x=−4
x equals negative 2 and x equals negative 4

x=2 and x=4

Answer 1

To solve the equation \((x+4)(x-2) = 0\), we set each factor equal to zero:

1. \(x + 4 = 0\) gives us \(x = -4\).
2. \(x - 2 = 0\) gives us \(x = 2\).

Therefore, the values of \(x\) that make the equation equal zero are \(x = -4\) and \(x = 2\).

The correct response is: x=−4 and x=2.

Answer 2

Solve the factored quadratic equation (19x−3)(9+x)=0.(1 point)
Responses

The solution set is x={−9,3}.
The solution set is x equals left brace negative 9 comma 3 right brace .

The solution set is x={−9,319}.
The solution set is x equals left brace negative 9 comma Start Fraction 3 over 19 End Fraction right brace .

The solution set is x={193,9}.
The solution set is x equals left brace Start Fraction 19 over 3 End Fraction comma 9 right brace .

The solution set is x={−9,193}.

Answer 3

To solve the equation \((19x - 3)(9 + x) = 0\), we set each factor equal to zero:

1. First Factor:
   \(19x - 3 = 0\)
   Adding 3 to both sides gives:
   \(19x = 3\)
   Dividing by 19 gives:
   \(x = \frac{3}{19}\)

2. Second Factor:
   \(9 + x = 0\)
   Subtracting 9 from both sides gives:
   \(x = -9\)

Therefore, the solution set is:
\({ -9, \frac{3}{19} }\).

The correct response is:
The solution set is x={−9, \(\frac{3}{19}\)}.

Answer 4

Solve the quadratic equation 84x2+16x=0 by factoring out the GCF.(1 point) Responses The solutions are x=−4 and x=−421. The solutions are x equals negative 4 and x equals negative Start Fraction 4 over 21 End Fraction . The solutions are x=−421 and x=0. The solutions are x equals negative Start Fraction 4 over 21 End Fraction and x equals 0 . The solutions are x=0 and x=4. The solutions are x equals 0 and x equals 4 . The solutions are x=0 and x=421.

Answer 5

To solve the quadratic equation \(84x^2 + 16x = 0\), we can factor out the greatest common factor (GCF) from both terms.

1. Find the GCF:
   The GCF of \(84x^2\) and \(16x\) is \(4x\).

2. Factor out the GCF:
   \(84x^2 + 16x = 4x(21x + 4) = 0\).

3. Set each factor equal to zero:
   \(4x = 0\) gives us:
   \(x = 0\).

   For the second factor:
   \(21x + 4 = 0\) gives us:
   \(21x = -4\)
   \(x = -\frac{4}{21}\).

The solutions to the equation are:
\(x = 0\) and \(x = -\frac{4}{21}\).

The correct response is:
The solutions are x=0 and x=−\(\frac{4}{21}\).

Answer 6

Which of the following quadratic equations is not solvable by grouping?(1 point)
Responses

2x2+14x+12=0
2 x squared plus 14 x plus 12 equals 0

x2−12x+35=0
x squared minus 12 x plus 35 equals 0

2x2−2x−10=0
2 x squared minus 2 x minus 10 equals 0

x2−2x+1=0
x squared minus 2 x plus 1 equals 0

Answer 7

To determine which of the given quadratic equations is not solvable by grouping, we can look for the factorable form of each equation. Grouping typically works well when the polynomial can be rearranged in a way that allows us to factor by grouping pairs.

1. For the equation \(2x^2 + 14x + 12 = 0\):

• This can be factored as \(2(x^2 + 7x + 6) = 0\), which can further be factored as \(2(x + 6)(x + 1) = 0\).
• This can be solved by grouping.

2. For the equation \(x^2 - 12x + 35 = 0\):

• This can be factored as \((x - 5)(x - 7) = 0\).
• This can be solved by grouping.

3. For the equation \(2x^2 - 2x - 10 = 0\):

• Factor out the GCF first: \(2(x^2 - x - 5) = 0\). The quadratic \(x^2 - x - 5\) does not have integer roots, and it doesn't neatly factor.
• This is less straightforward for grouping.

4. For the equation \(x^2 - 2x + 1 = 0\):

• This factors as \((x - 1)(x - 1) = (x - 1)^2 = 0\).
• This can be solved by grouping.

Conclusion

The equation \(2x^2 - 2x - 10 = 0\) is the one that is not easily solvable by grouping, since it does not factor neatly into a product of binomials.

Thus, the correct response is:
2x²−2x−10=0.

Answer 8

Jedida solved the quadratic equation x2−96=4x by factoring. Her work is shown below. At which step did Jedida first make a mistake, if at all?

x2−96=4x
Step 1: x2+4x−96=0 Add 4x to both sides.
Step 2: (x+12)(x−8)=0 Factor.
Step 3: x+12=0 or x−8=0 Use the Zero Product Property.
x=−12 or x=8
(1 point)
Responses

Step 3; she did not apply the Zero Product Property correctly.
Step 3; she did not apply the Zero Product Property correctly.

Step 2; she did not factor the left side of the equation correctly.
Step 2; she did not factor the left side of the equation correctly.

Step 1; she added 4x to both sides instead of subtracting.
Step 1; she added 4 x to both sides instead of subtracting.

She did not make any mistakes.

Answer 9

Let’s analyze each step of Jedida's process in solving the quadratic equation \(x^2 - 96 = 4x\).

1. Starting Equation: \[ x^2 - 96 = 4x \]

2. Step 1: \[ x^2 + 4x - 96 = 0 \] Here, she correctly added \(4x\) to both sides. This step is correct.

3. Step 2: \[ (x + 12)(x - 8) = 0 \] Let's verify the factoring. The product \( (x + 12)(x - 8) \) expands to: \[ x^2 - 8x + 12x - 96 = x^2 + 4x - 96 \] The factoring is correct. This step is also correct.

4. Step 3: \[ x + 12 = 0 \quad \text{or} \quad x - 8 = 0 \] Here, she is using the Zero Product Property correctly to set each factor equal to zero.

In summary:

• She did not make any mistakes in the steps shown.
• Her final solutions would be \(x = -12\) or \(x = 8\), which are correct based on her factoring.

Conclusion

The correct response is:
She did not make any mistakes.

Answer 10

Solve the following quadratic equation using the perfect square trinomial pattern: x2+x+0.25=0.(1 point)
Responses

x=0.5
x equals 0.5

x=−0.5
x equals negative 0.5

x=−0.25
x equals negative 0.25

x=0.25
x equals 0.25

Answer 11

To solve the quadratic equation \(x^2 + x + 0.25 = 0\) using the perfect square trinomial pattern, we need to recognize that the left side can be factored as a perfect square.

1. The quadratic equation can be rewritten as: \[ x^2 + x + 0.25 = 0 \]

2. Notice that \(0.25\) is equal to \(\left(\frac{1}{2}\right)^2\) and the middle term \(x\) can be expressed as \(2 \cdot x \cdot \frac{1}{2}\). Therefore, we can rewrite it as: \[ \left(x + \frac{1}{2}\right)^2 = 0 \]

3. This gives us: \[ \left(x + \frac{1}{2}\right)^2 = 0 \]

4. To solve for \(x\), take the square root of both sides: \[ x + \frac{1}{2} = 0 \] \[ x = -\frac{1}{2} \]

Thus, the solution of the quadratic equation is: \[ x = -0.5 \]

Conclusion

The correct response is:
x=−0.5.