Simplify this

Question

1. Original Equation: x
2
+4x−12=0
2. Factoring:
- He correctly factors the quadratic equation into (x+6)(x−2)=0.
3. Zero Product Property:
- Sets each factor equal to zero:
- x+6=0
- x−2=0
4. Solving:
- Solves the equations to find:
- x=−6
- x=2
Thus, Jamal's solution is accurate because he followed the correct steps to factor the quadratic equation and applied the zero product property correctly.
### George's Solution
1. Original Equation: x
2
+4x−5=7
2. Misstep:
- Did not correctly set the equation equal to zero. Instead, attempted to factor it directly.
- Incorrectly factors to (x+5)(x−1)=7 rather than equating all terms to zero.
3. Zero Product Property:
- Incorrectly applies the zero product property from (x+5)(x−1)=7, which isn't valid under this form.
4. Incorrect Solving:
- Solves (x+5)=0 and (x−1)=0 leading to:
- x=−5
- x=1
George’s solution is inaccurate because he did not set the quadratic equation equal to zero before factoring, which led to incorrect factors and solutions.
### Correct Approach for George's Equation
To solve George's equation correctly:
1. Set the Equation Equal to Zero:
- Start with x
2
+4x−5=7.
- Subtract 7 from both sides to obtain x
2
+4x−12=0.
2. Factoring:
- The correct factors for x
2
+4x−12 are (x+6)(x−2)=0.
3. Applying Zero Product Property:
- Set each factor to zero:
- x+6=0
- x−2=0
4. Solve:
- x=−6
- x=2
Thus, Jamal's original solution process was correct and could be applied accurately to both situations when done correctly from the start.

Answer 1

Simplified Comparison of Solutions

Jamal's Solution

1. Original Equation: \(x^2 + 4x - 12 = 0\)
2. Factoring: Correctly factors to \((x + 6)(x - 2) = 0\).
3. Zero Product Property: Sets each factor equal to zero:
   • \(x + 6 = 0\) → \(x = -6\)
   • \(x - 2 = 0\) → \(x = 2\)
4. Conclusion: Both solutions \(x = -6\) and \(x = 2\) are correct.

George's Solution

1. Original Equation: \(x^2 + 4x - 5 = 7\)
2. Misstep: Did not set the equation to zero; incorrectly attempted to factor.
   • Incorrectly factors \(x^2 + 4x - 5 - 7 = 0\) as \((x + 5)(x - 1) = 7\).
3. Zero Product Property: Incorrectly applies the zero product property:
   • Results in \((x + 5) = 0\) → \(x = -5\) and \((x - 1) = 0\) → \(x = 1\).
4. Conclusion: \(x = -5\) and \(x = 1\) are incorrect.

Correct Approach for George's Equation

1. Set to Zero: Start with \(x^2 + 4x - 5 = 7\).
   • Rearrange to \(x^2 + 4x - 12 = 0\) by subtracting 7 from both sides.
2. Factoring: Factor to \((x + 6)(x - 2) = 0\).
3. Apply Zero Product Property:
   • Set \(x + 6 = 0\) → \(x = -6\)
   • Set \(x - 2 = 0\) → \(x = 2\).
4. Conclusion: The correct solutions are \(x = -6\) and \(x = 2\).

Summary

• Jamal followed the correct steps to solve his equation successfully.
• George's initial method was flawed due to not setting the equation equal to zero, leading to incorrect factors and roots.
• By applying the correct approach, George could arrive at the same solutions as Jamal.