Solve the factored quadratic equation (19x−3)(x−9)=0

Question

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

The solution set is x={−9, −319}
𝑥
=
{
−
9
,
 
−
3
19
}
.
The solution set is   x={−9, −319}
𝑥
=
{
−
9
,
 
−
3
19
}
.

The solution set is x={3, 9}
𝑥
=
{
3
,
 
9
}
.
The solution set is    x={3, 9}
𝑥
=
{
3
,
 
9
}
.

The solution set is x={319, 9}
𝑥
=
{
3
19
,
 
9
}
.
The solution set is x={319, 9}
𝑥
=
{
3
19
,
 
9
}
.

The solution set is x={193, 9}
𝑥
=
{
19
3
,
 
9
}
.

GPT-4o mini · Answer

To solve the factored quadratic equation $(19x−3)(x−9)=0$, we can set each factor equal to zero.

Set the first factor to zero: \[ 19x - 3 = 0 \] \[ 19x = 3 \] \[ x = \frac{3}{19} \]
Set the second factor to zero: \[ x - 9 = 0 \] \[ x = 9 \]

So, the solutions to the equation are: \[ x = \frac{3}{19} \quad \text{and} \quad x = 9 \]

Thus, the solution set is: \[ x = \left{ \frac{3}{19}, 9 \right} \]

Based on the response options provided, the correct solution set is: The solution set is $ x = \left{ \frac{3}{19}, 9 \right} $ (though there may not be a perfectly matching option).

Yes. · Answer

Solve the quadratic equation 84x2+16x=0

by factoring out the GCF.(1 point)
Responses

The solutions are x=−421

The solutions are x=0 and x=421

The solutions are x=−4 and x=−421

The solutions are x=0 and x=4

GPT-4o mini · Answer

To solve the quadratic equation $84x^2 + 16x = 0$ by factoring out the greatest common factor (GCF), we first identify the GCF of the terms. The GCF of $84x^2$ and $16x$ is $4x$. Now, we can factor the equation: $$ 84x^2 + 16x = 4x(21x + 4) = 0 $$ Next, we'll set each factor equal to zero: 1. From the factor $4x = 0$: $$ x = 0 $$ 2. From the factor $21x + 4 = 0$: $$ 21x = -4 $$ $$ x = -\frac{4}{21} $$ Thus, the solutions to the equation $84x^2 + 16x = 0$ are: $$ x = 0 \quad \text{and} \quad x = -\frac{4}{21} $$ Given the response options provided, the correct solution set is: **The solutions are $x = 0$ and $x = -\frac{4}{21}$** (though this option doesn't appear exactly in your list). If $ -\frac{4}{21} $ is meant to resemble the format for one of the other options, you might clarify or check them again, as they do not match the calculated solutions.

Yes. · Answer

Find the mistake made when solving x2−96=4x
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

Responses:

Step 2 - She did not factor 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.

Step 1 - She did not set it =0, so she did not use the Zero Product Property correctly.

GPT-4o mini · Answer

Let's analyze the provided steps for solving the equation $ x^2 - 96 = 4x $.

Step 1: Rearranging the Equation

The original equation is: \[ x^2 - 96 = 4x \] The first step involves moving all terms to one side: \[ x^2 - 4x - 96 = 0 \] In this step, instead of adding $4x$ to both sides, it should have been subtracted from both sides to set the equation to zero correctly.

Step 2: Factoring

The equation $x^2 - 4x - 96 = 0$ can be factored as: \[ (x + 12)(x - 8) = 0 \] This step is correctly factored.

Step 3: Zero Product Property

Using the Zero Product Property correctly gives two potential solutions: \[ x + 12 = 0 \quad \Rightarrow \quad x = -12 \] \[ x - 8 = 0 \quad \Rightarrow \quad x = 8 \]

Conclusion

The mistake in the solution occurs in Step 1, where it was incorrectly stated that $4x$ was added to both sides instead of subtracted to bring it to the left side. Thus, the correct response is:

Step 1 - She added $4x$ to both sides instead of subtracting.

Yes. · Answer

Solve the following quadratic equation using the perfect square trinomial pattern: x2+x+0.25=0 0.25 Responses x=0.25 x=0.5 x=−0.25 x=−0.5

GPT-4o mini · Answer

To solve the quadratic equation $x^2 + x + 0.25 = 0$ using the perfect square trinomial pattern, we'll follow these steps: 1. **Recognize the perfect square trinomial:** The equation can be rewritten as: $$ x^2 + x + \left(\frac{1}{2}\right)^2 = 0 $$ Here, $\left(\frac{1}{2}\right)^2 = 0.25$. 2. **Rewrite the equation as a perfect square:** $$ (x + 0.5)^2 = 0 $$ 3. **Take the square root of both sides:** $$ x + 0.5 = 0 $$ 4. **Solve for $x$:** $$ x = -0.5 $$ Thus, the solution to the equation $x^2 + x + 0.25 = 0$ is: $$ \boxed{x = -0.5} $$ Based on the provided responses, the correct solution is: **$x = -0.5$**.

Yes. · Answer

Ginger was asked to solve 25x2−196=0 by factoring using the difference of squares pattern. What is/are the correct solution(s)? Responses x=196/25 x=14/5 x=−14/5, x=14/5

GPT-4o mini · Answer

To solve the equation $25x^2 - 196 = 0$ by factoring using the difference of squares pattern, we can follow these steps: 1. **Recognize the difference of squares:** The equation $25x^2 - 196$ can be expressed as a difference of squares: $$ (5x)^2 - (14)^2 = 0 $$ 2. **Apply the difference of squares formula:** The difference of squares can be factored as: $$ (5x - 14)(5x + 14) = 0 $$ 3. **Set each factor to zero:** We set each factor equal to zero: $$ 5x - 14 = 0 \quad \text{or} \quad 5x + 14 = 0 $$ - For $5x - 14 = 0$: $$ 5x = 14 \quad \Rightarrow \quad x = \frac{14}{5} $$ - For $5x + 14 = 0$: $$ 5x = -14 \quad \Rightarrow \quad x = -\frac{14}{5} $$ 4. **Write the solutions:** The solutions are: $$ x = \frac{14}{5} \quad \text{and} \quad x = -\frac{14}{5} $$ Based on the response options provided, the correct solution is: **$x = -\frac{14}{5}, x = \frac{14}{5}$**.

Yes. · Answer

Complete the square to solve the following quadratic equation: x2−6x−20=20 Responses x=10, x=−4 x=−10, x=4 x=52, x=−46

GPT-4o mini · Answer

To solve the quadratic equation $x^2 - 6x - 20 = 20$ by completing the square, follow these steps: 1. **Simplify the equation:** Start by moving $20$ to the left side: $$ x^2 - 6x - 20 - 20 = 0 $$ $$ x^2 - 6x - 40 = 0 $$ 2. **Isolate the quadratic and linear terms:** Add $40$ to both sides: $$ x^2 - 6x = 40 $$ 3. **Complete the square:** To complete the square, take half of the coefficient of $x$ (which is $-6$), square it, and add it to both sides. Half of $-6$ is $-3$, and squaring it gives $9$. Add $9$ to both sides: $$ x^2 - 6x + 9 = 40 + 9 $$ $$ (x - 3)^2 = 49 $$ 4. **Take the square root of both sides:** $$ x - 3 = \pm 7 $$ 5. **Solve for $x$:** - For $x - 3 = 7$: $$ x = 10 $$ - For $x - 3 = -7$: $$ x = -4 $$ Thus, the solutions to the equation $x^2 - 6x - 20 = 20$ are: $$ x = 10, \quad x = -4 $$ Based on the response options provided, the correct solution is: **$x = 10, x = -4$**.