mmon Factors Practice

Question

mmon Factors Practice
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Question
Solve the following quadratic equation by factoring out the greatest common factor (GCF):  2x2−14x=0
 .(1 point)
The smaller solution is x=
, and the larger solution is x=
.
Remaining Attempts : 3

GPT-4o mini · Answer

To solve the quadratic equation $2x^2 - 14x = 0$ by factoring out the greatest common factor (GCF), we first identify the GCF of the terms in the equation. 1. The GCF of $2x^2$ and $-14x$ is $2x$. 2. We can factor out $2x$ from the equation: $$ 2x(x - 7) = 0 $$ 3. Now, we set each factor equal to zero to find the solutions: $$ 2x = 0 \quad \text{or} \quad x - 7 = 0 $$ 4. Solving these equations: - For $2x = 0$: $$ x = 0 $$ - For $x - 7 = 0$: $$ x = 7 $$ Thus, the smaller solution is $x = 0$, and the larger solution is $x = 7$.

adaleta · Answer

Rewrite the following quadratic equation in standard form and then solve by factoring out the GCF: 6x2=22x .(1 point) The smaller solution is x= , and the larger solution is x=

GPT-4o mini · Answer

To rewrite the equation $6x^2 = 22x$ in standard form and then solve by factoring out the greatest common factor (GCF), follow these steps: 1. **Move all terms to one side of the equation:** $$ 6x^2 - 22x = 0 $$ 2. **Identify the GCF of the terms $6x^2$ and $-22x$.** The GCF is $2x$. 3. **Factor out the GCF:** $$ 2x(3x - 11) = 0 $$ 4. **Set each factor equal to zero to find the solutions:** $$ 2x = 0 \quad \text{or} \quad 3x - 11 = 0 $$ 5. **Solve these equations:** - For $2x = 0$: $$ x = 0 $$ - For $3x - 11 = 0$: $$ 3x = 11 \quad \Rightarrow \quad x = \frac{11}{3} $$ Thus, the smaller solution is $x = 0$, and the larger solution is $x = \frac{11}{3}$.

adaleta · 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)=−x2+12x
, whereh(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.

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:

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

Rearranging the equation gives:

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

Factoring out the common term:

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

Setting each factor to zero gives us:
- For $-x = 0$: \[ x = 0 \]
- For $x - 12 = 0$: \[ x = 12 \]

This means the balloon will reach the ground at $x = 12$ feet.

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

adaleta · Answer

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
.

GPT-4o mini · Answer

Let's analyze Karim's steps in solving the quadratic equation $x^2 + 5x - 24 = 0$ by grouping.

Given Steps:

Step 1: $x^2 + 8x + (-3x) - 24 = 0$
Step 2: $(x^2 + 8x) + (-3x - 24) = 0$
Step 3: $x(x + 8) + 3(x + 8) = 0$
Step 4: $(x + 8)(x + 3) = 0$
Step 5: $x + 8 = 0$ or $x + 3 = 0$

Identify the Error:

In Step 1, Karim rewrote $5x$ incorrectly as $8x + (-3x)$. The correct factors of $-24$ that add to $5$ are $8$ and $-3$, so Step 1 is incorrect. The equation should be written as:

\[ x^2 + 8x - 3x - 24 = 0 \]
This continues on incorrectly in the next steps.

Conclusion:

Karim's first mistake occurs at Step 1.

adaleta · Answer

it was 3

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

GPT-4o mini · Answer

To solve the quadratic equation $x^2 - 13x + 42 = 0$ using the method of splitting the middle term, we need to find two numbers that multiply to the constant term ($42$) and add up to the coefficient of the middle term ($-13$). 1. We need two numbers that: - Multiply to $42$ (the constant term). - Add up to $-13$ (the coefficient of $x$). 2. The pairs of factors of $42$ are: - $1$ and $42$ - $2$ and $21$ - $3$ and $14$ - $6$ and $7$ 3. Out of these pairs, we need to consider the negative signs to make the sum equal to $-13$: - $-6$ and $-7$ 4. Checking their sum: - $-6 + (-7) = -13$ - $-6 \times -7 = 42$ Thus, the two numbers needed to complete the step are $-6$ and $-7$. ### Answer: - First number: $-7$ - Second number: $-6$ So, you would enter: $-7$ and $-6$.