Solve the following quadratic equation by factoring out the greatest common factor (GCF): 2x^2−14x=0

Question

Solve the following quadratic equation by factoring out the greatest common factor (GCF): 2x^2−14x=0
.(1 point)
The smaller solution is x= (    )
, and the larger solution is x= (     )

GPT-4o mini · Answer

To solve the quadratic equation $2x^2 - 14x = 0$, we can start by factoring out the greatest common factor (GCF). 1. Identify the GCF of the terms $2x^2$ and $-14x$. The GCF is $2x$. 2. Factor out the GCF from the equation: $$ 2x(x - 7) = 0 $$ 3. Set each factor equal to zero: $$ 2x = 0 \quad \text{or} \quad x - 7 = 0 $$ 4. Solve for $x$: - From $2x = 0$: $$ x = 0 $$ - From $x - 7 = 0$: $$ x = 7 $$ Thus, the solutions to the equation are $x = 0$ and $x = 7$. The smaller solution is $x = 0$ and the larger solution is $x = 7$. So, the final answers are: - The smaller solution is $x = 0$, and the larger solution is $x = 7$.

x · Answer

Rewrite the following quadratic equation in standard form and then solve by factoring out the GCF: 6x^2=22x .(1 point) The smaller solution is x= ( ) , and the larger solution is x= ( )

GPT-4o mini · Answer

To rewrite the quadratic equation $6x^2 = 22x$ in standard form, we first move all terms to one side of the equation: $$ 6x^2 - 22x = 0 $$ Next, we can factor out the greatest common factor (GCF) from the left side of the equation. The GCF of $6x^2$ and $-22x$ is $2x$. We factor it out: $$ 2x(3x - 11) = 0 $$ Next, we set each factor equal to zero: 1. From $2x = 0$: $$ x = 0 $$ 2. From $3x - 11 = 0$: $$ 3x = 11 \quad \Rightarrow \quad x = \frac{11}{3} $$ Now we have two solutions: - The smaller solution is $x = 0$ - The larger solution is $x = \frac{11}{3}$ So, the final answers are: - The smaller solution is $x = 0$, and the larger solution is $x = \frac{11}{3}$.

x · Answer

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)=−x^2+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=−x^2+12x
. How far will the balloon travel before it reaches the ground?(1 point)
The balloon will travel ( ) feet before it reaches the ground.

GPT-4o mini · Answer

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

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

Step 1: Rearranging the Equation

First, rearranging gives us:

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

Step 2: Factoring the Equation

Next, we can factor out the greatest common factor (GCF), which is $ -x $:

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

Step 3: Setting Each Factor to Zero

Now, we can set each factor equal to zero:

$ -x = 0 $ \[ x = 0 \]
$ x - 12 = 0 $ \[ x = 12 \]

Step 4: Interpreting the Results

The solutions we obtained are $ x = 0 $ and $ x = 12 $. The $ x = 0 $ solution represents the point where the balloon is fired, while $ x = 12 $ represents the point where the balloon reaches the ground.

Conclusion

Thus, the distance the balloon will travel before it reaches the ground is 12 feet.

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

x · Answer

Karim is solving the quadratic equation x^2+5x−24=0
by grouping. His work is shown below. Determine the first step at which Karim makes an error.

Step 1: x^2+8x+(−3x)−24=0

Step 2: (x62+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

GPT-4o mini · Answer

Let's analyze each step of Karim's work to determine where he makes an error.

Given equation: \[ x^2 + 5x - 24 = 0 \]

Step 1:
\[ x^2 + 8x + (-3x) - 24 = 0 \]
Here, Karim incorrectly splits $5x$ into $8x - 3x$. The correct representation should be: \[ x^2 + 5x - 24 = x^2 + 8x - 3x - 24 = 0 \]

Step 2:
\[ (x^2 + 8x) + (-3x - 24) = 0 \]
This step keeps the error from Step 1, but the grouping is consistent with those incorrect terms.

Step 3:
\[ x(x + 8) + 3(x + 8) = 0 \]
From the incorrect factorization due to the mistake in Step 1, Karim’s expressions are not accurate.

Step 4:
\[ (x + 8)(x + 3) = 0 \]
This step implies that the previous factors were correctly derived. However, they were based on the incorrect grouping in Step 1.

Step 5:
Setting each factor to zero is the correct mathematical procedure.

Conclusion

Karim's first mistake occurs at Step 1, where he incorrectly split $5x$ into $8x - 3x$ instead of correctly factoring or grouping. The correct grouping or factoring should have preserved the original components that conform to the standard quadratic expressions.

x · Answer

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 x^2−13x+42=0
, which two numbers would be needed to complete this step? Enter the lesser value first.(1 point)
x^2−13x+42=0
x2+ ( )x+ ( )x+42=0

GPT-4o mini · Answer

To solve the quadratic equation $x^2 - 13x + 42 = 0$ by grouping, we need to split the middle term $-13x$ into two terms that add up to $-13$ and multiply to $42$. 1. We need two numbers that add up to $-13$ and multiply to $42$. 2. The factors of $42$ are: - $1, 42$ - $2, 21$ - $3, 14$ - $6, 7$ Now, we need to find a pair of these factors that adds up to $-13$. - If we take $-6$ and $-7$: - $-6 + (-7) = -13$ - $-6 \times -7 = 42$ Thus, the two numbers to split $-13x$ are $-6$ and $-7$. Now we can express the equation in the form: $$ x^2 - 6x - 7x + 42 = 0 $$ ### Final Answer The two values needed to complete the step are: **-7** (lesser value) and **-6** (greater value): So, you would enter: **( -7 )** and **( -6 )**.