Question

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

Let's solve the quadratic equation given by \( x^2 + 4x - 5 = 7 \) step-by-step before analyzing Jamal's and George's solutions.

1. Move all terms to one side: \[ x^2 + 4x - 5 - 7 = 0 \ x^2 + 4x - 12 = 0 \]

2. Now, we want to factor the quadratic expression \( x^2 + 4x - 12 \). To factor this, we need two numbers that multiply to \(-12\) (the constant term) and add to \(4\) (the coefficient of the linear term). The numbers \(6\) and \(-2\) meet these criteria. So, we can factor as follows: \[ (x + 6)(x - 2) = 0 \]

3. Using the Zero Product Property, we set each factor equal to zero: \[ x + 6 = 0 \quad \Rightarrow \quad x = -6 \ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]

4. The solutions to the equation are: \[ x = -6 \quad \text{and} \quad x = 2 \]

Evaluation of Solutions

Now, we can evaluate the solutions provided by Jamal and George. Assuming each used the Zero Product Property after step 2, we can analyze their reasoning based on the correctness of their factoring and subsequent solutions.

1. Jamal's work:

   • If Jamal correctly factored \( x^2 + 4x - 12 \) as \( (x + 6)(x - 2) \) and obtained \( x = -6 \) and \( x = 2 \), then his work is accurate.

2. George's work:

   • If George made a mistake in factoring (for example, if he factored it incorrectlly, such as \( (x + 3)(x + 4) \)), he would derive a different set of roots and thus have inaccurate solutions.

Identifying the Correct Solution

To determine whose solution is correct, you would check the factorizations and the resulting roots:

• For the correct solution:

  • Check that \( (x + 6) \) and \( (x - 2) \) indeed multiply to \( x^2 + 4x - 12 \).
  • Verify that each solution satisfies the original equation.

• For the incorrect solution:

  • If the product does not yield \( x^2 + 4x - 12 \) upon expansion, or if substituting the incorrect solutions into the original equation does not satisfy it, then the solution is wrong.

Conclusion

The accurate solutions are confirmed by correctly applying the Zero Product Property to the factored quadratic \( (x + 6)(x - 2) = 0 \), yielding \( x = -6 \) and \( x = 2 \). Incorrect solutions arise from errors in factoring or misapplication of solving methods, leading to roots that do not satisfy the original equation.