Solve the quadratic equation 5x^2+75x=0 by factoring out the GCF.(1 point)

Question

Responses

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

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

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.

Answer 1

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

The GCF of \(5x^2\) and \(75x\) is \(5x\).

Now, we can factor out \(5x\) from the equation:

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

Now, we set each factor equal to zero to solve for \(x\):

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

Thus, the solutions to the equation are:

• \(x = 0\)
• \(x = -15\)

The correct response is: The solutions are \(x = -15\) and \(x = 0\).

Answer 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 path h(x)=−x^2+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=−x^2+15x . How far will the firework travel before it reaches the ground?(1 point)
Responses

15 feet
15 feet

−15 feet
negative 15 feet

7.5 feet
7.5 feet

30 feet
30 feet

Answer 3

To determine how far the firework will travel before it reaches the ground, we need to solve the quadratic equation given by setting \( h(x) = 0 \):

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

We can factor out the common term:

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

Setting each factor equal to zero gives us:

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

The solution \( x = 0 \) represents the point when the firework is launched, and \( x = 15 \) represents the horizontal distance traveled when the firework reaches the ground.

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

Answer 4

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 5

Let's go through Hyung-eun's work step by step to identify the first error.

The original equation is:

\[ x^2 - 4x - 12 = 0 \]

Step 1: Hyung-eun rewrites the equation as: \[ x^2 + (-6x) + 2x - 12 = 0 \] This step appears to be intended to break the middle term \(-4x\) into \(-6x\) and \(2x\). However, \(-6x + 2x\) equals \(-4x\), which is correct. So, no error here.

Step 2: In this step, she groups the first two terms and the last two terms: \[ (x^2 + (-6x)) + (2x - 12) = 0 \] This grouping is also correct, so no error here as well.

Step 3: Then, she factors each group: \[ x(x + (-6)) + 2(x - 6) = 0 \] Here, she appears to have made a mistake. The correct factorization of \(2x - 12\) is \(2(x - 6)\), which is correct. However, she wrote \(x + (-6)\) instead of \(x - 6\).

The expression \((x + (-6))\) should just be \((x - 6)\).

Step 4: She writes: \[ x(x - 6) + 2(x - 6) = 0 \] This is indeed correct because it can be factored again correctly.

Step 5: Finally, she factors further: \[ (x - 6)(x + 2) = 0 \] This should be correct if Step 3 had been accurate, but since it was incorrect, the result in this step is also improper.

Conclusion: The first error occurs in Step 3, where she incorrectly used \((x + (-6))\) instead of \((x - 6)\).

The correct answer is: Step 3.

Answer 6

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

x^2+10x+21=0
x squared plus 10 x plus 21 equals 0

x^2−4x−8=0
x squared minus 4 x minus 8 equals 0

x^2+8x−22=0
x squared plus 8 x minus 22 equals 0

x^2−12x+18=0

Answer 7

To determine which quadratic equations can be solved by grouping, we need to look for equations where the quadratic can be expressed in a form that allows us to factor it easily. This generally involves finding two numbers that multiply to give the constant term (the product) and add to give the middle coefficient (the sum).

Let's analyze each of the given equations:

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

   • Factors of \(21\) that add up to \(10\) are \(3\) and \(7\) (since \(3 + 7 = 10\)).
   • This can be factored as \((x + 3)(x + 7) = 0\).
   • Can be solved by grouping.

2. \(x^2 - 4x - 8 = 0\):

   • We need two numbers that multiply to \(-8\) and add to \(-4\). No integer factor pairs work for \(-8\) to yield \(-4\) (e.g., \(-2 + 4\) or \(-1 + 8\)), so this does not factor nicely using simple integers.
   • Cannot be easily solved by grouping.

3. \(x^2 + 8x - 22 = 0\):

   • We need two numbers that multiply to \(-22\) and add to \(8\). The pairs of integers that work are \(11\) and \(-2\) (since \(11 - 2 = 9\)).
   • This can be evaluated but might not yield nice integers.
   • Cannot be easily solved by grouping.

4. \(x^2 - 12x + 18 = 0\):

   • We need two numbers that multiply to \(18\) and add to \(-12\). The integer pair \(-6\) and \(-6\) work, allowing it to factor as \((x - 6)(x - 3)\).
   • Can be solved by grouping.

Conclusion: The equations that can be solved by grouping are:

• \(x^2 + 10x + 21 = 0\)
• \(x^2 - 12x + 18 = 0\)

Each of these equations can be factored easily by finding appropriate integer pairs.

If only one option is to be chosen, based on the common requirement that a quadratic should give two distinct integer solutions, the best answer would be: \(x^2 + 10x + 21=0\).

Answer 8

Photosynthesis occurs in a plant cell’s(1 point)
Responses

chloroplasts.
chloroplasts.

cell wall.
cell wall.

nucleus.
nucleus.

ribosomes.
ribosomes.

Answer 9

Photosynthesis occurs in a plant cell’s chloroplasts.

Chloroplasts contain chlorophyll, the pigment that captures light energy from the sun and converts it into chemical energy through the process of photosynthesis.

So the correct response is: chloroplasts.

Answer 10

Which option is a light wavelength that chlorophyll does not absorb?(1 point)
Responses

red
red

violet
violet

yellow
yellow

green

Answer 11

Chlorophyll primarily absorbs light in the blue (around 430 nm) and red (around 660 nm) parts of the spectrum. It does not absorb well in the green part of the spectrum (around 500-550 nm), which is why chlorophyll appears green to our eyes.

Thus, the light wavelength that chlorophyll does not absorb is green.

So the correct response is: green.