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.

Answer 1

To determine which student's solution is accurate, we need to analyze their work step-by-step.

Jamal's Solution

1. Step 1: Jamal sets the equation to 0: \[ x^2 + 4x - 12 = 0 \] This is incorrect because the original equation \(x^2 + 4x - 5 = 7\) should be manipulated to bring all terms to one side by subtracting 7 from both sides, yielding: \[ x^2 + 4x - 12 = 0. \]
   (This step is valid, so he did correctly set the equation to 0).

2. Step 2: He factors the equation: \[ (x + 6)(x - 2) = 0 \] This factorization is incorrect because when we expand \((x + 6)(x - 2)\), we get: \[ x^2 + 4x - 12. \] Therefore, his factors do not represent the correct quadratic equation.

3. Step 3: Jamal sets each factor to 0 and solves: \[ x + 6 = 0 \quad \Rightarrow \quad x = -6 \] \[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \] Since the factorization is incorrect, the solutions he obtained are also incorrect.

George's Solution

1. Step 1: George starts with the original equation: \[ x^2 + 4x - 5 = 7 \] He should have brought all terms to one side by subtracting 7, which would give: \[ x^2 + 4x - 12 = 0. \] But he appears to mistakenly start from \(x^2 + 4x - 5 = 7\) rather than bringing 7 to the left side.

2. Step 2: He factors the equation: \[ (x + 5)(x - 1) = 7 \] This step is also flawed because both \(x + 5\) and \(x - 1\) do not equal 7. The correct factorization of \(x^2 + 4x - 12 = 0\) should lead to something that equals to zero rather than 7.

3. Step 3: He sets each factor equal to zero: \[ x + 5 = 0 \quad \Rightarrow \quad x = -5 \] \[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \] Since his factorization and earlier manipulations were incorrect, these solutions cannot be verified as valid.

Conclusion

Both solutions contain inaccuracies, but we can see the breakdown:

• Jamal's work is fundamentally flawed in the factorization step: His factors \((x + 6)(x - 2) = 0\) do not reflect the correct quadratic form derived from the equation. Hence, his solutions \(x = -6\) and \(x = 2\) are incorrect.

• George starts correctly with the original equation but fails to express it in the correct factored form and incorrectly suggests \(x^2 + 4x - 12 = 7\) leading to wrong factors. Thus, \(x = -5\) and \(x = 1\) are also incorrect.

Since neither student provided the proper steps to derive the correct solutions for the equation as set, there is no accurate solution between them. The correct method involves bringing the equation to zero correctly and correctly factoring it into the right expressions to use the Zero Product Property.

To conclude, both students indeed made mistakes while solving, and any conclusion on their correctness cannot be made until the accurate factorization and resolution of \(x^2 + 4x - 12 = 0\) is done, leading to valid solutions.

Answer 2

can you make the whole thing ten times smaller

Answer 3

Sure! Here’s a more concise version:

Jamal's Solution

1. Step 1: Jamal correctly sets the equation to 0: \[ x^2 + 4x - 12 = 0. \]

2. Step 2: He factors it incorrectly as: \[ (x + 6)(x - 2) = 0. \] This is wrong; the correct factorization is different.

3. Step 3: \[ x + 6 = 0 \quad \Rightarrow \quad x = -6, \] \[ x - 2 = 0 \quad \Rightarrow \quad x = 2. \] Both solutions are incorrect due to the wrong factorization.

George's Solution

1. Step 1: George starts from \(x^2 + 4x - 5 = 7\) but fails to set it to 0 correctly.

2. Step 2: He factors as: \[ (x + 5)(x - 1) = 7, \] which is incorrect.

3. Step 3: \[ x + 5 = 0 \quad \Rightarrow \quad x = -5, \] \[ x - 1 = 0 \quad \Rightarrow \quad x = 1. \] Both solutions are also incorrect.

Conclusion

Both students made mistakes: Jamal's factors are wrong, and George’s factorization leads them astray. Neither solution is accurate, as both failed to apply the Zero Product Property correctly on the intended quadratic.

Answer 4

shorter

Answer 5

Jamal's Solution:

1. Sets the equation correctly to \( x^2 + 4x - 12 = 0 \).
2. Incorrectly factors as \( (x + 6)(x - 2) = 0 \), leading to wrong solutions \( x = -6 \) and \( x = 2 \).

George's Solution:

1. Starts incorrectly but should arrive at \( x^2 + 4x - 12 = 0 \).
2. Incorrectly factors as \( (x + 5)(x - 1) = 7 \), leading to wrong solutions \( x = -5 \) and \( x = 1 \).

Conclusion: Neither student correctly applied the Zero Product Property, leading to incorrect solutions. Both made critical errors in their factorization.

Answer 6

a little shorter

Answer 7

Jamal's Solution:

1. Sets the equation to \( x^2 + 4x - 12 = 0 \).
2. Incorrectly factors it as \( (x + 6)(x - 2) = 0 \), giving wrong solutions \( x = -6 \) and \( x = 2 \).

George's Solution:

1. Missteps in setting up the equation but ends up with \( x^2 + 4x - 12 = 0 \).
2. Incorrectly factors it as \( (x + 5)(x - 1) = 7 \), yielding wrong solutions \( x = -5 \) and \( x = 1 \).

Conclusion: Both students made errors in factorization and applied the Zero Product Property incorrectly, resulting in incorrect solutions.