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

Question

GPT-4o mini · Accepted Answer

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 two terms, which is $ 5x $. So, we can factor the equation as follows: $$ 5x(x + 15) = 0 $$ Now, we set each factor equal to zero: 1. $ 5x = 0 $ - Dividing both sides by 5 gives: $ x = 0 $ 2. $ x + 15 = 0 $ - Solving for $ x $ gives: $ x = -15 $ Thus, the solutions to the equation $ 5x^2 + 75x = 0 $ are: $$ x = 0 \quad \text{and} \quad x = -15 $$ From the options provided, the correct answer is: **The solutions are $ x = -15 $ and $ x = 0 $.**

Finally Some Answers · Answer

Which of the following tables shows the correct steps to factor out the GCF and solve the quadratic equation 7x^2=56x ?(1 point)

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=−8 and x=0
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=−7 and x=8
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
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

GPT-4o mini · Answer

To factor and solve the quadratic equation $ 7x^2 = 56x $, we first need to rewrite it in standard form. This can be done by moving all terms to one side of the equation, which gives us:

\[ 7x^2 - 56x = 0 \]

Next, we find the greatest common factor (GCF), which in this case is $ 7x $. We factor that out:

\[ 7x(x - 8) = 0 \]

Now, we set each factor equal to zero:

$ 7x = 0 $
- Dividing both sides by 7 gives: $ x = 0 $
$ x - 8 = 0 $
- Solving for $ x $ gives: $ x = 8 $

The solutions to the equation $ 7x^2 - 56x = 0 $ are:

\[ x = 0 \quad \text{and} \quad x = 8 \]

Now, looking at the provided options, the correct table of steps is:

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 $

Finally Some Answers · Answer

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 30 feet 30 feet 7.5 feet

GPT-4o mini · Answer

To solve the quadratic equation $ 0 = -x^2 + 15x $ to determine how far the firework will travel before it reaches the ground, we can first rewrite it in standard form: $$ -x^2 + 15x = 0 $$ Now, we can factor out the greatest common factor, which is $ -x $: $$ -x(x - 15) = 0 $$ Setting each factor equal to zero gives us: 1. $ -x = 0 $ - This leads to $ x = 0 $ 2. $ x - 15 = 0 $ - This leads to $ x = 15 $ The solutions are $ x = 0 $ and $ x = 15 $. The height $ h(x) = 0 $ occurs as the firework is either launched or has landed. Since we are interested in how far it travels before it reaches the ground, the answer is: $$ \text{The firework will travel } 15 \text{ feet.} $$ Thus, the correct response is: **15 feet.**

Finally Some Answers · Answer

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 1 Step 1 Step 4 Step 4 Step 3 Step 3 Step 2 Step 2 Skip to navigation

GPT-4o mini · Answer

Let's go through Hyung-eun's steps to identify where the error occurs in her process of solving the quadratic equation $ x^2 - 4x - 12 = 0 $ by grouping.

Step 1: $ x^2 + (-6x) + 2x - 12 = 0 $
- This step attempts to split the middle term $-4x$ into $-6x$ and $2x$. This actually may look like it could work, but the correct split of $-4x$ into two terms that sum to $-4$ and multiply to $-12$ should actually be $-6x + 2x$. So she could have thought she was on the right track, but the arithmetical combinations were incorrectly assigned since $-6 + 2 = -4$ but doesn't accurately reflect the proper factors.
Step 2: $ (x^2 + (-6x)) + (2x - 12) = 0 $
- This step accurately groups the two parts but can be supported with the understanding that assigning the split terms correctly lays a clearer pathway forward.
Step 3: $ x(x + (-6)) + 2(x - 6) = 0 $
- Here she attempts to factor each group; however, the factor should be $ x(x - 6) + 2(x - 6) = 0 $. She has a sign mistake in interpreting her junction with the added terms, causing a fundamental issue.
Step 4: $ (x + 6)(x - 2) = 0 $
- This is where the mistake propagates. Based on her prior compositions, the correct factorization after the summations should lead to $ (x - 6)(x + 2) $.
Step 5: $ x + 6 = 0 $ or $ x - 2 = 0 $
- She finds $ x = -6 $ and $ x = 2 $ based on incorrectly factored equations.

So, the first error that Hyung-eun makes clearly occurs in Step 3, where she misinterpreted the correct grouping leading to misfactoring.

Thus, the correct response is:

Step 3.

Finally Some Answers · Answer

Which of the following quadratic equations can be solved by grouping?(1 point) x^2+8x−22=0 x^2−12x+18=0 x^2−4x−8=0 x^2+10x+21=0

GPT-4o mini · Answer

To determine which of the given quadratic equations can be solved by grouping, we typically check if we can find two numbers that multiply to the product of the coefficient of $x^2$ (which is $1$ in all these equations) and the constant term, while adding up to the coefficient of the linear term ($x$).

The general form of a quadratic equation is $ax^2 + bx + c = 0$. In this case, $a = 1$ for all equations.

1. $ x^2 + 8x - 22 = 0 $

Here, the product $ac = 1 \cdot (-22) = -22$ and $b = 8$.
We need two numbers that multiply to $-22$ and add to $8$.
The factors $10$ and $-2$ satisfy this condition: $10 \cdot (-2) = -20$, not viable.
This cannot be solved by grouping.

2. $ x^2 - 12x + 18 = 0 $

The product $ac = 1 \cdot 18 = 18$ and $b = -12$.
We need factors that multiply to $18$ and add to $-12$.
The numbers are $-6$ and $-6$ (since $-6 + (-6) = -12$); however, they don't fit grouping directly. Nevertheless, this format generally supports grouping.

3. $ x^2 - 4x - 8 = 0 $

The product $ac = 1 \cdot (-8) = -8$ and $b = -4$.
We need to find two numbers that multiply to $-8$ and add to $-4$.
The pairs (2 and -4) do not fit thusly.

4. $ x^2 + 10x + 21 = 0 $

The product $ac = 1 \cdot 21 = 21$ and $b = 10$.
We need two numbers that multiply to $21$ and add to $10$.
The numbers $3$ and $7$ do meet this condition: $3 + 7 = 10$.

Now, summing this up, the second and fourth equations can potentially be solved by grouping, giving emphasis in this context particularly to $ x^2 - 12x + 18 = 0 $ and $ x^2 + 10x + 21 = 0 $.

Among these options, the quadratic equations that can truly be solved by grouping based on clear identifiable successful pair factorization are primarily $ x^2 - 12x + 18 = 0 $ and $ x^2 + 10x + 21 = 0 $.

In conclusion, the one more likely achievable cleanly through the grouping process is:

$ x^2 + 10x + 21 = 0 $.

Respectively, it has more fluid access through concepts of factoring by grouping.