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

Question

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 expression. The terms $2x^2$ and $-14x$ have a GCF of $2x$. We factor out $2x$ from the equation: $$ 2x(x - 7) = 0 $$ Now, we can set each factor equal to zero: 1. $2x = 0$ 2. $x - 7 = 0$ Solving the first equation: $$ 2x = 0 \implies x = 0 $$ Solving the second equation: $$ x - 7 = 0 \implies x = 7 $$ Thus, the solutions to the equation are $x = 0$ and $x = 7$. The smaller solution is $x = 0$, and the larger solution is $x = 7$. Final answers: - The smaller solution is $x = 0$. - The larger solution is $x = 7$.

Honeybeee · 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=blank, and the larger solution is x=blank.

GPT-4o mini · Answer

To rewrite the equation $6x^2 = 22x$ in standard form, we first move all terms to one side of the equation: $$ 6x^2 - 22x = 0 $$ Next, we identify the greatest common factor (GCF) of the terms $6x^2$ and $-22x$. The GCF is $2x$. We factor this out: $$ 2x(3x - 11) = 0 $$ Now, we can set each factor equal to zero: 1. $2x = 0$ 2. $3x - 11 = 0$ Solving the first equation: $$ 2x = 0 \implies x = 0 $$ Solving the second equation: $$ 3x - 11 = 0 \implies 3x = 11 \implies x = \frac{11}{3} $$ Thus, the solutions to the equation are $x = 0$ and $x = \frac{11}{3}$. Final answers: - The smaller solution is $x = 0$. - The larger solution is $x = \frac{11}{3}$.

Honeybeee · 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 , 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 blank 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 will set the height $h(x)$ equal to zero and solve the quadratic equation:

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

First, we can rewrite the equation:

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

Next, we factor out the greatest common factor:

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

Setting each factor equal to zero:

$-x = 0$
$x - 12 = 0$

Solving the first equation:

\[ -x = 0 \implies x = 0 \]

Solving the second equation:

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

The solutions indicate that the balloon is at ground level when $x = 0$ (the point of launch) and when $x = 12$ feet.

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

Final answer:

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

Honeybeee · 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 blank.

GPT-4o mini · Answer

Let's analyze each of Karim's steps to find the first mistake:

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

In this step, Karim is trying to split the middle term $5x$ into two parts. The correct way to split $5x$ is into $8x$ and $-3x$, since $8 + (-3) = 5$. This step is okay, even though breaking $5x$ into $8x$ and $-3x$ feels unconventional.
Step 2: $(x^2 + 8x) + (-3x - 24) = 0$

This step is also correct; he is grouping the terms correctly now.
Step 3: $x(x + 8) + 3(x + 8) = 0$

Here, he factors out $x$ from $x^2 + 8x$ correctly to get $x(x + 8)$, but he mistakenly factors out $3$ from $-3x - 24$. The correct factorization should be $-3(x + 8)$, not $+3(x + 8)$, because $-3x - 24$ can be factored as $-3(x + 8)$.
Step 4: $(x + 8)(x + 3) = 0$

Because of the mistake made in Step 3, this step is incorrect.
Step 5: $x + 8 = 0$ or $x + 3 = 0$, which leads to $x = -8$ or $x = -3$

This conclusion is also incorrect due to the earlier mistake.

Thus, Karim's first mistake occurs at Step 3.

Final answer:

Karim's first mistake occurs at Step 3.

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

GPT-4o mini · Answer

To split the middle term of the quadratic equation $x^2 - 13x + 42 = 0$, we need to find two numbers that multiply to the constant term $42$ and add up to the coefficient of the middle term $-13$.

Let's denote the two numbers as $m$ and $n$. We need:

$m + n = -13$ (the coefficient of $x$)
$m \cdot n = 42$ (the constant term)

We can find the pairs of factors of $42$:

$1 \times 42$
$2 \times 21$
$3 \times 14$
$6 \times 7$

Considering the sum $m + n = -13$, both numbers must be negative because two negative numbers will add to a negative sum. The pairs of negative factors are:

$-1$ and $-42$
$-2$ and $-21$
$-3$ and $-14$
$-6$ and $-7$

Now, let’s determine which pair adds up to $-13$:

$-2 + (-21) = -23$
$-3 + (-14) = -17$
$-6 + (-7) = -13$ (this pair works)

So, the two numbers we need to split the middle term are $-6$ and $-7$.

Now we can rewrite the equation as:

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

In terms of blanks:

The answer is:

The first blank: -6
The second blank: -7

Final answer:

$x^2 + \mathbf{-6}x + \mathbf{-7}x + 42 = 0$