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)

Answer 1

To solve the quadratic equation \( x^2 + 4x - 5 = 7 \), we first need to rewrite it in standard form. We do this by moving all the terms to one side:

\[ x^2 + 4x - 5 - 7 = 0 \implies x^2 + 4x - 12 = 0 \]

Now, we can factor this quadratic expression. We are looking for two numbers that multiply to \(-12\) (the constant term) and add to \(4\) (the coefficient of the \(x\) term). The pair that works is \(6\) and \(-2\), so we can factor the equation:

\[ (x + 6)(x - 2) = 0 \]

Now we can use the Zero Product Property, which tells us that if a product of factors equals zero, at least one of the factors must be zero:

1. \(x + 6 = 0\) gives us \(x = -6\)
2. \(x - 2 = 0\) gives us \(x = 2\)

Thus, the solutions to the equation \( x^2 + 4x - 12 = 0 \) are \( x = -6 \) and \( x = 2 \).

Evaluating the Student Solutions

If Jamal and George provided different interpretations or factorizations, we should evaluate their steps to determine which one is correct.

Indicators of Correctness:

• The correct solution should factor the equation correctly to \((x + 6)(x - 2) = 0\).
• It should identify both solutions as \( x = -6 \) and \( x = 2 \).

Indicators of Inaccuracy:

• An error in factoring (e.g., misidentifying the factors or incorrectly applying the Zero Product Property) will lead to incorrect solutions.
• For example, if one student had a factorization such as \((x + 4)(x - 3)\), it would be clear that they made an error because their factors would not yield the correct original quadratic equation or the right values for \(x\).

Conclusion

To determine definitively which student's solution is accurate, you must check their factorization and the derived solutions against the criteria we've established. The correct student will have correctly factored the quadratic and successfully applied the Zero Product Property, resulting in the correct solutions of \( x = -6 \) and \( x = 2 \). The incorrect student's solution will either have the wrong factors or will give erroneous values for \(x\), demonstrating a misunderstanding of the process.