percy solved the equation $x^2 + 7x + 12 = 12$. his work is shown below. is percy correct? explain.\n1. $(x…

Explanation:

Step 1: Analyze the original equation

The original equation is (x^{2}+7x + 12=12). First, we should simplify this equation. Subtract 12 from both sides: (x^{2}+7x+12 - 12=12 - 12) (x^{2}+7x=0)

Step 2: Analyze Percy's first step

Percy factored the left - hand side as ((x + 3)(x + 4)=12). But when we use the zero - product property (if (ab = 0), then (a = 0) or (b = 0)), the right - hand side of the equation must be zero. In Percy's equation ((x + 3)(x + 4)=12), the right - hand side is 12 (not zero), so we cannot directly set (x + 3=12) or (x + 4=12) using the zero - product property.

Step 3: Solve the equation correctly

Starting from (x^{2}+7x+12 = 12), subtract 12 from both sides to get (x^{2}+7x=0). Then factor out an (x): (x(x + 7)=0). Using the zero - product property, we have (x=0) or (x+7 = 0). If (x+7 = 0), then (x=-7).

Answer:

Percy is not correct. The error occurs in Step 2. The zero - product property (if (ab = c), we can only set (a = 0) or (b = 0) when (c = 0)). In the equation ((x + 3)(x + 4)=12), the right - hand side is 12 (not zero), so we cannot set (x + 3 = 12) or (x + 4 = 12) directly. The correct solution is obtained by first subtracting 12 from both sides of the original equation to get (x^{2}+7x=0), then factoring as (x(x + 7)=0), and using the zero - product property to find (x = 0) or (x=-7).