Rewrite the following quadratic equation in standard form and then solve by factoring out the GCF: 6x2=22x

To rewrite the given quadratic equation \( 6x^2 = 22x \) in standard form, we need to move all terms to one side of the equation:

\[ 6x^2 - 22x = 0 \]

Next, we can factor out the greatest common factor (GCF), which in this case is \( 2x \):

\[ 2x(3x - 11) = 0 \]

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

1. \( 2x = 0 \)

   • Dividing both sides by 2 gives: \[ x = 0 \]

2. \( 3x - 11 = 0 \)

   • Adding 11 to both sides gives: \[ 3x = 11 \]
   • Dividing both sides by 3 gives: \[ x = \frac{11}{3} \]

Now, we have both solutions:

• The smaller solution is \( x = 0 \)
• The larger solution is \( x = \frac{11}{3} \)

Final answers:

• The smaller solution is \( x = 0 \)
• The larger solution is \( x = \frac{11}{3} \)

During a water balloon fight, Louisa uses a slingshot to fire a water balloon over the fortified wall of the opponent. When fired from the slingshot, the balloon will follow the path h(x)=−x2+12x
, where h(x)
is the height of the balloon in feet, and x is the horizontal distance it will travel. To determine how far the balloon will travel before it reaches the ground, Louisa sets the height equal to zero and solves the quadratic equation 0=−x2+12x
. How far will the balloon travel before it reaches the ground?(1 point)
The balloon will travel
feet before it reaches the ground.

To determine how far the balloon will travel before it reaches the ground, we need to solve the quadratic equation given by the height of the balloon set equal to zero:

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

First, we can rearrange the equation:

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

Next, we can factor out the GCF, which is \( -x \):

\[ -x(x - 12) = 0 \]

Now, we can set each factor equal to zero:

1. \( -x = 0 \)

   • This gives: \[ x = 0 \]

2. \( x - 12 = 0 \)

   • This gives: \[ x = 12 \]

The two solutions are \( x = 0 \) (when the balloon is initially launched) and \( x = 12 \) (when it hits the ground).

Therefore, the balloon will travel 12 feet before it reaches the ground.

Final answer:

• The balloon will travel 12 feet before it reaches the ground.

Karim is solving the quadratic equation x2+5x−24=0
by grouping. His work is shown below. Determine the first step at which Karim makes an error.

Step 1: x2+8x+(−3x)−24=0


Step 2: (x2+8x)+(−3x−24)=0


Step 3: x(x+8)+3(x+8)=0


Step 4: (x+8)(x+3)=0


Step 5: x+8=0
or x+3=0


x=−8
or x=−3


(1 point)
Karim's first mistake occurs at Step
.

Let’s analyze Karim's steps to identify where the error occurs:

1. Step 1: \( x^2 + 8x + (-3x) - 24 = 0 \)

   Here, Karim splits the middle term \( 5x \) incorrectly. The correct ways to factor \( 5x \) into two terms such that they multiply to \(-24\) (the constant term) should be \(8x\) and \(-3x\), which is not correct for this equation since these two terms do not add up to \(5x\). The correct split of \(5x\) into two terms that sum to \(5\) and multiply to \(-24\) should actually be \(8x\) and \(-3x\). This is the first mistake.

2. Step 2: \( (x^2 + 8x) + (-3x - 24) = 0 \)

   This grouping is based on Step 1 and doesn’t correct the previous mistake.

3. Step 3: \( x(x + 8) + 3(x + 8) = 0 \)

   This step is based on the incorrect grouping including the error made in Step 1.

4. Step 4: \( (x + 8)(x + 3) = 0 \)

   This step remains based on the incorrect factorization.

5. Step 5: Solving the factors gives Karim the two possible solutions: \( x + 8 = 0 \) or \( x + 3 = 0 \), leading to the answers \( x = -8 \) or \( x = -3 \).

Thus, the first mistake occurs in Step 1, where the middle term is incorrectly split.

Final Answer: Karim's first mistake occurs at Step 1.

One of the steps needed to solve a quadratic equation by grouping is to split the middle term as a sum of two terms. Given the equation x2−13x+42=0
, which two numbers would be needed to complete this step? Enter the lesser value first.(1 point)
x2−13x+42=0

x2+
x+
x+42=0

To solve the quadratic equation \( x^2 - 13x + 42 = 0 \) by grouping, we need to find two numbers that:

1. Add up to \(-13\) (the coefficient of the linear term).
2. Multiply to \(42\) (the constant term).

Let’s find two numbers that satisfy these conditions.

The pairs of factors of \(42\) are:

• \(1\) and \(42\)
• \(2\) and \(21\)
• \(3\) and \(14\)
• \(6\) and \(7\)

Now we examine which of these pairs can add up to \(-13\) when both numbers are negatives:

• \(-1\) and \(-42\) add to \(-43\)
• \(-2\) and \(-21\) add to \(-23\)
• \(-3\) and \(-14\) add to \(-17\)
• \(-6\) and \(-7\) add to \(-13\) ✔️

The two numbers that work are \(-6\) and \(-7\).

Thus, you can rewrite the equation as:

\[ x^2 - 6x - 7x + 42 = 0 \]

Therefore, the solution to your query is:

Answer: \(-7\) and \(-6\) (entering the lesser value first).

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

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 .

The solutions are x=0
and x=5
.

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

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

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

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

Next, we set each factor equal to zero:

1. \(5x = 0\)

   • Dividing both sides by \(5\) gives: \[ x = 0 \]

2. \(x + 15 = 0\)

   • Subtracting \(15\) from both sides gives: \[ x = -15 \]

The solutions to the equation are:

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

Thus, the correct response is:

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

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 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=−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 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=0
and x=8

To solve the quadratic equation \(7x^2 = 56x\) by factoring out the GCF, let's go through the appropriate steps:

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

2. Factor out the GCF: The GCF of \(7x^2\) and \(56x\) is \(7x\). \[ 7x(x - 8) = 0 \]

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

4. Solve for \(x\):

   • From \(7x = 0\): \[ x = 0 \]
   • From \(x - 8 = 0\): \[ x = 8 \]

Thus, the solutions are: \[ x = 0 \quad \text{and} \quad x = 8 \]

Now, let's evaluate which of the provided choices matches these steps correctly.

The correct choice 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 \(x = 8\)

This is reflected in the following choice:

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.

Thus, this is the correct and complete set of steps for factoring out the GCF and solving the quadratic equation \(7x^2 = 56x\).