Use the tables to answer the question.

Question

Jamal
Step 1: Set equation =0.	x2+4x−12=0
Step 2: Write quadratic equation as the product of two factors.	(x+6)(x−2)=0
Step 3: Set each factor equal to 0.	x+6=0 and x−2=0
Step 4: Solve each equation.	x=−6 and x=2
George
Step 1: Begin with the equation.	x2+4x−5=7
Step 2: Write quadratic equation as the product of two factors.	(x+5)(x−1)=7
Step 3: Set each factor equal to 0.	x+5=0 and x−1=0
Step 4: Solve each equation.	x=−5 and x=1
When trying to solve the quadratic equation x2+4x−5=7, Jamal and George each provided their work in the tables. Each said they used the Zero Product Property to solve the equation after step 2. Explain how you know the correct student’s solution is accurate and how you know the incorrect student’s solution is inaccurate.

(2 points)
 
Short Answer Rubric (2 points)
Points Awarded	Criteria
2	
The answer is very clear.

The answer is fully supported by details.

There may be a few errors in grammar, usage, or mechanics. Errors do not interfere with meaning.

1	
The answer is somewhat clear.

Details are weak or not related.

There are some errors in grammar, usage, and mechanics. Errors interfere with meaning.

0	The question is not answered.

9th grade level

GPT-4o mini · Answer

To determine which student's solution is accurate, we need to evaluate the steps taken by both Jamal and George.

Jamal started with the equation $ x^2 + 4x - 12 = 0 $. When he factored it as $ (x + 6)(x - 2) = 0 $, he correctly set each factor equal to zero, giving the solutions $ x = -6 $ and $ x = 2 $. However, we need to check if his original quadratic equation is equivalent to the one posed, which was actually $ x^2 + 4x - 5 = 7 $. To solve this, we subtract 7 from both sides to transform it into standard form: $ x^2 + 4x - 12 = 0 $. Therefore, Jamal's steps are mathematically correct for the equation after transformation.

On the other hand, George began with $ x^2 + 4x - 5 = 7 $ but incorrectly wrote it as $ (x + 5)(x - 1) = 7 $. This factorization is incorrect. Instead, the correct factorization should yield $ (x + 6)(x - 2) = 0 $ when we bring everything to one side. Subsequently, George's factors do not lead to valid solutions for the original equation. His approach is flawed because he misapplied the factoring step.

In summary, the correct student’s solution is accurate because he factored and solved the equation properly once it was set to zero, and he adhered to the Zero Product Property correctly. George's solution is inaccurate because he misfactored the equation, resulting in incorrect solutions and failing to recognize the proper relationship between the original equation and its factors.

adaleta · Answer

one parigraph

GPT-4o mini · Answer

Jamal's solution is accurate while George's is not because of their respective approaches to the quadratic equation. Jamal correctly transformed the equation $ x^2 + 4x - 5 = 7 $ into standard form as $ x^2 + 4x - 12 = 0 $ and then accurately factored it as $ (x + 6)(x - 2) = 0 $. By applying the Zero Product Property, he set each factor to zero and found the correct solutions, $ x = -6 $ and $ x = 2 $. In contrast, George incorrectly factored the equation as $ (x + 5)(x - 1) = 7 $, which is not valid. His error in the factoring step led to incorrect conclusions about the roots. Therefore, Jamal's method is supported by correct mathematical principles, while George's approach is flawed.

adaleta · Answer

Determine the zero(s) of the quadratic function y=6x2+13x+5. Show all steps.(2 points)

Short Answer Rubric (2 points)
Points Awarded Criteria
2
The answer is very clear.

The answer is fully supported by details.

There may be a few errors in grammar, usage, or mechanics. Errors do not interfere with meaning.

1
The answer is somewhat clear.

Details are weak or not related.

There are some errors in grammar, usage, and mechanics. Errors interfere with meaning.

0 The question is not answered.

GPT-4o mini · Answer

To determine the zero(s) of the quadratic function $ y = 6x^2 + 13x + 5 $, we start by setting the function equal to zero:

Set the equation to zero: \[ 6x^2 + 13x + 5 = 0 \]
Factor the quadratic equation: First, we need to find two numbers that multiply to $ 6 \times 5 = 30 $ (the product of the leading coefficient and the constant term) and add up to $ 13 $ (the coefficient of the middle term). The numbers that satisfy this are $ 10 $ and $ 3 $.

Next, we rewrite the middle term using these numbers: \[ 6x^2 + 10x + 3x + 5 = 0 \]

Now, we can group the terms: \[ (6x^2 + 10x) + (3x + 5) = 0 \] Factor out the common factors from each group: \[ 2x(3x + 5) + 1(3x + 5) = 0 \] Now factor out $ (3x + 5) $: \[ (2x + 1)(3x + 5) = 0 \]
Set each factor equal to zero: \[ 2x + 1 = 0 \quad \text{and} \quad 3x + 5 = 0 \]
Solve for $ x $: For the first equation $ 2x + 1 = 0 $: \[ 2x = -1 \quad \Rightarrow \quad x = -\frac{1}{2} \]

For the second equation $ 3x + 5 = 0 $: \[ 3x = -5 \quad \Rightarrow \quad x = -\frac{5}{3} \]
State the zeros: The zeros of the quadratic function $ y = 6x^2 + 13x + 5 $ are $ x = -\frac{1}{2} $ and $ x = -\frac{5}{3} $.

Thus, the zeros are clearly determined as $ \boxed{-\frac{1}{2}} $ and $ \boxed{-\frac{5}{3}} $.

adaleta · Answer

one paragraph

GPT-4o mini · Answer

To determine the zeros of the quadratic function $ y = 6x^2 + 13x + 5 $, we start by setting the equation equal to zero: $ 6x^2 + 13x + 5 = 0 $. Next, we seek two numbers that multiply to $ 30 $ (the product of the leading coefficient $ 6 $ and the constant $ 5 $) and add up to $ 13 $. The numbers $ 10 $ and $ 3 $ meet these criteria. We rewrite the equation as $ 6x^2 + 10x + 3x + 5 = 0 $ and then factor it by grouping: $ (2x + 1)(3x + 5) = 0 $. Setting each factor equal to zero, we solve $ 2x + 1 = 0 $ to find $ x = -\frac{1}{2} $ and $ 3x + 5 = 0 $ to find $ x = -\frac{5}{3} $. Therefore, the zeros of the function are $ x = -\frac{1}{2} $ and $ x = -\frac{5}{3} $.

adaleta · Answer

Complete the square to determine the vertex and axis of symmetry of y=−x2+8x+7.(4 points)

Short Answer Rubric (4 points)
Points Awarded Criteria
4
The answer is very clear.

The answer is fully supported by details.

There may be a few errors in grammar, usage, or mechanics. Errors do not interfere with meaning.

3
The answer is mostly clear.

The answer is supported by mostly relevant details.

There are a few errors in grammar, usage, or mechanics. Errors do not interfere with meaning.

2
The answer is somewhat clear.

Details are weak or not related.

There are some errors in grammar, usage, and mechanics. Errors interfere with meaning.

1
The answer is not clear.

There are few or no supporting details.

There are many errors in grammar, usage, and mechanics. Errors interfere with meaning.

0 The question is not answered.

one paragraph

GPT-4o mini · Answer

To determine the vertex and axis of symmetry of the quadratic function $ y = -x^2 + 8x + 7 $ by completing the square, we first rewrite the function in vertex form. We start with the expression $ -x^2 + 8x $ and factor out the negative sign: $ -(x^2 - 8x) $. Next, we complete the square inside the parentheses. To do this, we take half of the coefficient of $ x $, which is $ -8 $, giving us $ -4 $, and square it to get $ 16 $. We then add and subtract $ 16 $ inside the parentheses: $ -((x^2 - 8x + 16) - 16) + 7 $. This can be rewritten as $ -(x - 4)^2 + 16 + 7 $, or $ - (x - 4)^2 + 23 $. Thus, the vertex of the parabola is at the point $ (4, 23) $, and the axis of symmetry is the vertical line $ x = 4 $. Therefore, the vertex is $ (4, 23) $ and the axis of symmetry is $ x = 4 $.