Question

Which of the following tables shows the correct steps to factor out the GCF and solve the quadratic equation 7x2=56x?(1 point)
Responses

Put the quadratic in standard form x−8=0
Factor out the GCF No GCF
Set each factor equal to zero x−8=0
The solutions are x=8 Put the quadratic in standard form x minus 8 equals 0 Factor out the GCF No GCF Set each factor equal to zero x minus 8 equals 0 The solutions are x equals 8

Put the quadratic in standard form 7x2+56x=0
Factor out the GCF 7x(x+8)=0
Set each factor equal to zero 7x=0 and x+8=0
The solutions are x=−8 and x=0 Put the quadratic in standard form 7 x squared plus 56 x equals 0 Factor out the GCF 7 x left parenthesis x plus 8 right parenthesis equals 0 Set each factor equal to zero 7 x equals 0 and x plus 8 equals 0 The solutions are x equals negative 8 and x equals 0

Put the quadratic in standard form 7x2−56x=0
Factor out the GCF 7x(x−8)=0
Set each factor equal to zero 7x=0 and x−8=0
The solutions are x=0 and x=8Put the quadratic in standard form 7 x squared minus 56 x equals 0 Factor out the GCF 7 x left parenthesis x minus 8 right parenthesis equals 0 Set each factor equal to zero 7 x equals 0 and x minus 8 equals 0 The solutions are x equals 0 and x equals 8

Put the quadratic in standard form 7x2−56x=0
Factor out the GCF 7x(x−8)=0
Set each factor equal to zero 7x=0 and x−8=0
The solutions are x=−7 and x=8
2. As Isla prepares to set off fireworks, she wants to ensure the spectators are a safe distance away. When shot in the air, one firework will follow the pathh(x)=−x2+15x, where h(x) is the height of the firework, and x is the horizontal distance it travels in feet. To determine how far the firework will travel before it reaches the ground, Isla sets the height equal to zero, and solves the following quadratic equation: 0=−x2+15x. How far will the firework travel before it reaches the ground?(1 point)
Responses

15 feet
15 feet

7.5 feet
7.5 feet

30 feet
30 feet
−15 feet
3. Hyung-eun is solving the quadratic equation x2−4x−12=0 by grouping. Her work is shown below. At which step does Hyung-eun first make an error?

x2−4x−12=0

Step1: x2+(−6x)+2x−12=0

Step 2: (x2+(−6x))+(2x−12)=0

Step 3: x(x+(−6))+2(x−6)=0

x(x−6)+2(x−6)=0

Step 4: (x+6)(x−2)=0

Step 5: x+6=0 or x−2=0

x=−6 or x=2

(1 point)
Responses

Step 2
Step 2

Step 1
Step 1

Step 4
Step 4

Step 3
4.
Which of the following quadratic equations can be solved by grouping?(1 point)
Responses

x2+8x−22=0
x squared plus 8 x minus 22 equals 0

x2−4x−8=0
x squared minus 4 x minus 8 equals 0

x2+10x+21=0
x squared plus 10 x plus 21 equals 0

x2−12x+18=0
x squared minus 12 x plus 18 equals 0

Answer 1

Let's address each question in turn:

1. **Factoring the Quadratic Equation**:
The original equation is \(7x^2 = 56x\). We want to bring it into standard form, then factor it out. The correct steps are:

**Put the quadratic in standard form**:
\[
7x^2 - 56x = 0
\]
**Factor out the GCF**:
\[
7x(x - 8) = 0
\]
**Set each factor equal to zero**:
\[
7x = 0 \quad \text{and} \quad x - 8 = 0
\]
**The solutions are**:
\[
x = 0 \quad \text{and} \quad x = 8
\]

Therefore, the correct response is:
**Put the quadratic in standard form 7x²−56x=0
Factor out the GCF 7x(x−8)=0
Set each factor equal to zero 7x=0 and x−8=0
The solutions are x=0 and x=8**.

2. **Firework Trajectory**:
The equation given is \(0 = -x^2 + 15x\). To determine how far the firework travels before it reaches the ground, you can solve for \(x\):

Rearranged, it gives:
\[
x^2 - 15x = 0
\]
Factoring out \(x\):
\[
x(x - 15) = 0
\]
Setting each factor to zero gives:
\[
x = 0 \quad \text{or} \quad x = 15
\]

Therefore, the firework will travel **15 feet** before it reaches the ground. The correct response is: **15 feet**.

3. **Hyung-eun's Error**:
Let's analyze the steps.

- **Step 1**: \(x^2 - 4x - 12 = 0\)
- **Step 2**: \(x^2 + (-6x) + 2x - 12 = 0\)
- **Step 3**: Grouping correctly seems to proceed to separate terms.
- **Step 4**: The factorization leading to \((x + 6)(x - 2) = 0\) does not yield the original quadratic.

The first apparent error occurs at **Step 1**, because the way \(x^2 - 4x - 12\) is grouped is incorrect. The correct grouping would be splitting the middle term into -6 and +2 correctly before factoring.

Thus, the answer is: **Step 1**.

4. **Which Quadratic Can Be Solved by Grouping?**
To solve by grouping, the quadratic must be factored into two binomials. The key to remember is that we need to find two numbers that multiply to \(a \cdot c\) (the product of the coefficient of \(x^2\) and the constant term) and that add to \(b\) (the coefficient of \(x\)).

Let's take a look at each option:

- **x² + 8x - 22 = 0**: Cannot be grouped easily.
- **x² - 4x - 8 = 0**: Hard to group.
- **x² + 10x + 21 = 0**: Factors easily into \((x + 3)(x + 7)\), which is not grouping.
- **x² - 12x + 18 = 0**: Factors can lead to useful pairs but difficult.

The one that clearly could be grouped with integer factors would be \(x^2 - 12x + 18 = 0\) since it could potentially lead to a couple of usable values directly leading to a binomial solution.

Thus, the best answer is **x² - 12x + 18 = 0** can be solved by grouping.

Hope that helps! If you have further questions or need clarification, feel free to ask!