Question 1 A)Solve the following quadratic equation by factoring out the greatest common factor (GCF): 2x2−14x=0.(1 point) The smaller solution is x=$$, and the larger solution is x=$$. Question 2 A)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=$$, and the larger solution is x=$$. Question 3 A)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 $$ feet before it reaches the ground. Question 4 A) Karim is solving the quadratic equation x2+5x−24=0 by grouping. His work is shown below. Determine the first step at which Karim makes an error. Step 1: x2+8x+(−3x)−24=0 Step 2: (x2+8x)+(−3x−24)=0 Step 3: x(x+8)+3(x+8)=0 Step 4: (x+8)(x+3)=0 Step 5: x+8=0 or x+3=0 x=−8 or x=−3 (1 point) Karim's first mistake occurs at Step $$. Question 5 A)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

Question

GPT-4o mini · Answer

Let's go through each of the questions one by one:

Question 1

To solve the quadratic equation $2x^2 - 14x = 0$, we first factor out the greatest common factor (GCF):

The GCF of $2x^2$ and $-14x$ is $2x$.
Factor out $2x$: \[ 2x(x - 7) = 0 \]
Set each factor equal to zero: \[ 2x = 0 \quad \Rightarrow \quad x = 0 \] \[ x - 7 = 0 \quad \Rightarrow \quad x = 7 \]

The smaller solution is $x = 0$, and the larger solution is $x = 7$.

Question 2

To rewrite the equation $6x^2 = 22x$ in standard form, we need to move all terms to one side:

Rearrange the equation: \[ 6x^2 - 22x = 0 \]
Factor out the GCF, which is $2x$: \[ 2x(3x - 11) = 0 \]
Set each factor equal to zero: \[ 2x = 0 \quad \Rightarrow \quad x = 0 \] \[ 3x - 11 = 0 \quad \Rightarrow \quad 3x = 11 \quad \Rightarrow \quad x = \frac{11}{3} \]

The smaller solution is $x = 0$, and the larger solution is $x = \frac{11}{3}$.

Question 3

For the path of the water balloon given by the equation $h(x) = -x^2 + 12x$, we set it to zero to find when it hits the ground:

Set the height equal to zero: \[ 0 = -x^2 + 12x \]
Factor out the GCF: \[ 0 = -x(x - 12) \]
Set each factor equal to zero: \[ -x = 0 \quad \Rightarrow \quad x = 0 \] \[ x - 12 = 0 \quad \Rightarrow \quad x = 12 \]

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

Question 4

Karim is solving the equation $x^2 + 5x - 24 = 0$.

Step 1: $x^2 + 8x - 3x - 24 = 0$ - Here, the attempt to split the middle term is wrong because $8$ and $-3$ do not add to $5$. The correct split would be to find two numbers that multiply to $-24$ and add to $5$.

Karim's first mistake occurs at Step 1.

Question 5

To split the middle term in the equation $x^2 - 13x + 42 = 0$, we need two numbers that multiply to $42$ and add to $-13$:

The two numbers are $-6$ and $-7$ since:

$-6 \times -7 = 42$
$-6 + -7 = -13$

Thus, we rewrite the equation as: \[ x^2 - 6x - 7x + 42 = 0 \]

So, the two numbers are $-6$ and $-7$.

The answers to the problems are:

The smaller solution is $0$, and the larger solution is $7$.
The smaller solution is $0$, and the larger solution is $\frac{11}{3}$.
The balloon will travel $12$ feet before it reaches the ground.
Karim's first mistake occurs at Step $1$.
$x^2 - 6x - 7x + 42 = 0$.