Solve the quadratic equation 5x2+75x=0 by factoring out the GCF.(1 point)

Question

Solve the quadratic equation 5x2+75x=0 by factoring out the GCF.(1 point)

Responses

The solutions are x=0 and x=5.
The solutions are x equals 0 and x equals 5 .

The solutions are x=0 and x=15.
The solutions are x equals 0 and x equals 15 .

The solutions are x=−15 and x=0.
The solutions are x equals negative 15 and x equals 0 .

The solutions are x=−15 and x=−5.
The solutions are x equals negative 15 and x equals negative 5 .

👇

9 answers
(click or scroll down)

👇

Answer 1

Bot GPT 4-o mini answered
24 days ago

To solve the quadratic equation \(5x^2 + 75x = 0\) by factoring out the greatest common factor (GCF), we first identify the GCF of the terms.

The GCF of \(5x^2\) and \(75x\) is \(5x\). Now, we can factor out \(5x\) from the equation:

\[
5x(x + 15) = 0
\]

Next, we set each factor equal to zero:

1. \(5x = 0\) which gives \(x = 0\)
2. \(x + 15 = 0\) which gives \(x = -15\)

Therefore, the solutions to the equation are:

\[
x = 0 \quad \text{and} \quad x = -15
\]

Thus, the correct response is:

**The solutions are \(x = -15\) and \(x = 0\).**

Answer 2

anonymous answered
24 days ago

Which of the following tables shows the correct steps to factor out the GCF and solve the quadratic equation 7x^2=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=−7 and x=8
Put 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 negative 7 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=0 and x=8
Put 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=−8 and x=0

Answer 3

Bot GPT 4-o mini answered
24 days ago

Let's solve the quadratic equation \(7x^2 = 56x\).

1. **Put the quadratic in standard form:**
\[
7x^2 - 56x = 0
\]

2. **Factor out the GCF (Greatest Common Factor):**
The GCF of \(7x^2\) and \(56x\) is \(7x\), so we factor it out:
\[
7x(x - 8) = 0
\]

3. **Set each factor equal to zero:**
\[
7x = 0 \quad \text{and} \quad x - 8 = 0
\]

4. **Solve for \(x\):**
- From \(7x = 0\), we get \(x = 0\).
- From \(x - 8 = 0\), we get \(x = 8\).

So the solutions are \(x = 0\) and \(x = 8\).

Now, let's find the response that matches these steps:

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

The correct response is:

**Put 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.**

Answer 4

anonymous answered
24 days ago

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 path h(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

30 feet
30 feet

−15 feet
negative 15 feet

15 feet
15 feet

7.5 feet

Answer 5

Bot GPT 4-o mini answered
24 days ago

To determine how far the firework will travel before it reaches the ground, we need to solve the equation:

\[
0 = -x^2 + 15x
\]

First, we can rearrange it for clarity:

\[
-x^2 + 15x = 0
\]

Next, we can factor out the common terms:

\[
x(-x + 15) = 0
\]

Setting each factor equal to zero gives us:

1. \(x = 0\)
2. \(-x + 15 = 0\) → \(x = 15\)

Thus, the firework will travel a total of \(15\) feet before it reaches the ground.

Therefore, the correct response is:

**15 feet.**

Answer 6

anonymous answered
24 days ago

Hyung-eun is solving the quadratic equation x^2−4x−12=0 by grouping. Her work is shown below. At which step does Hyung-eun first make an error?

x^2−4x−12=0

Step1: x^2+(−6x)+2x−12=0

Step 2: (x^2+(−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 4
Step 4

Step 3
Step 3

Step 2
Step 2

Step 1

Answer 7

Bot GPT 4-o mini answered
24 days ago

Let's analyze Hyung-eun's work step by step to identify where she made an error in solving the quadratic equation \(x^2 - 4x - 12 = 0\) by grouping.

1. **Original equation:**
\[
x^2 - 4x - 12 = 0
\]

2. **Step 1:**
\[
x^2 + (-6x) + 2x - 12 = 0
\]
- This step is **incorrect.** The equation should be split into two terms that actually sum to \(-4x\) and still multiply to \(-12\). The correct pairs of factors for \(-12\) that sum to \(-4\) are actually \(-6\) and \(2\), but Hyung-eun incorrectly placed the signs or selected terms.

3. **Step 2:**
\[
(x^2 + (-6x)) + (2x - 12) = 0
\]
- Since Step 1 is already incorrect, the grouping in Step 2 is based on an incorrect assumption from Step 1.

4. **Step 3:**
\[
x(x + (-6)) + 2(x - 6) = 0
\]
- Incorrect because the grouping is not valid to the original equation.

5. **Step 4:**
\[
(x + 6)(x - 2) = 0
\]
- This expression is produced incorrectly since Steps 2 and 3 were built on a mistake.

6. **Step 5:**
\[
x + 6 = 0 \quad \text{or} \quad x - 2 = 0
\]
- Solving these roots doesn't matter because they are based on incorrect factors.

Given this analysis, Hyung-eun first makes an error in **Step 1** when she incorrectly sets up the grouping.

Therefore, the correct response is:

**Step 1.**

Answer 8

anonymous answered
24 days ago

Which of the following quadratic equations can be solved by grouping?(1 point)

Responses

x2−12x+18=0
x squared minus 12 x plus 18 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+8x−22=0

Answer 9

Bot GPT 4-o mini answered
24 days ago

To determine which quadratic equations can be solved by grouping, we need to find the ones that can be factored into two binomials. Generally, for this method to work, the middle coefficient (the coefficient of \(x\)) needs to be able to be expressed as the sum of two numbers that multiply to give the product of the coefficient of \(x^2\) (which is 1 in all these cases) and the constant term.

Let's analyze each equation:

1. **\(x^2 - 12x + 18 = 0\)**
- Here, we need two numbers that add up to \(-12\) and multiply to \(18\).
- The numbers \(-6\) and \(-6\) do not satisfy this because they multiply to \(36\).
- **Cannot be solved by grouping.**

2. **\(x^2 - 4x - 8 = 0\)**
- We need numbers that add to \(-4\) and multiply to \(-8\).
- The numbers \(-6\) and \(2\) add to \(-4\) and multiply to \(-12\).
- **Cannot be solved by grouping.**

3. **\(x^2 + 10x + 21 = 0\)**
- We need numbers that add to \(10\) and multiply to \(21\).
- The numbers \(3\) and \(7\) fit this since \(3 + 7 = 10\) and \(3 \cdot 7 = 21\).
- This can be factored as \((x + 3)(x + 7) = 0\).
- **Can be solved by grouping (after initial factorization).**

4. **\(x^2 + 8x - 22 = 0\)**
- We need numbers that add to \(8\) and multiply to \(-22\).
- The numbers \(11\) and \(-2\) fit this since \(11 - 2 = 9\) and the product is \(-22\).
- This can be factored as \((x + 11)(x - 2) = 0\).
- **Can be solved by grouping (after initial factorization).**

The only quadratic equation that can be solved by grouping (after performing the necessary factorization) is:

**\(x^2 + 10x + 21 = 0\)**.