Percy has made some mistakes in solving the equation [tex]\(x^2 + 7x + 12 = 12\)[/tex]. Let's go through the correct steps to solve this equation and highlight where Percy went wrong.
### Step-by-Step Solution
1. Start with the given equation:
[tex]\[ x^2 + 7x + 12 = 12 \][/tex]
2. Simplify the equation by subtracting 12 from both sides to set it equal to zero:
[tex]\[ x^2 + 7x + 12 - 12 = 0 \][/tex]
[tex]\[ x^2 + 7x = 0 \][/tex]
3. Now factor the quadratic equation [tex]\(x^2 + 7x = 0\)[/tex]:
To factor this, we can take out the common factor [tex]\(x\)[/tex]:
[tex]\[ x(x + 7) = 0 \][/tex]
4. Set each factor to zero to find the solutions:
[tex]\[ x = 0 \][/tex]
[tex]\[ x + 7 = 0 \][/tex]
[tex]\[ x = -7 \][/tex]
Thus, the correct solutions to the equation [tex]\( x^2 + 7x + 12 = 12 \)[/tex] are:
[tex]\[ x = 0 \text{ and } x = -7 \][/tex]
### Identifying Percy's Mistakes
- Percy's first mistake:
He incorrectly factored the equation. He wrote [tex]\( (x + 3)(x + 4) = 12 \)[/tex] instead of simplifying the original equation to [tex]\( x^2 + 7x = 0 \)[/tex].
- Percy's second mistake:
After his incorrect factorization, he set each factor equal to 12, i.e., [tex]\( x + 3 = 12 \)[/tex] or [tex]\( x + 4 = 12 \)[/tex]. This step is also incorrect. The correct approach after factorization would have been to set each factor to zero.
- Percy's third mistake:
Based on his incorrect second step, he found [tex]\( x = 9 \)[/tex] and [tex]\( x = 8 \)[/tex], which are not solutions to the given equation.
Therefore, Percy's solutions [tex]\( x = 9 \)[/tex] and [tex]\( x = 8 \)[/tex] are incorrect.
The correct solutions are:
[tex]\[ \boxed{0 \text{ and } -7} \][/tex]
