Find the Domain of y = -(x+7)² + 12: Complete Analysis

To solve this problem, we follow these essential steps:

• Step 1: Set $y = 0$ to find the roots of the equation.
• Step 2: Analyze the expression $-\left(x+7\right)^2 + 12 = 0$.
• Step 3: Isolate $\left(x+7\right)^2$ by adding $-12$ to both sides, so: $\left(x+7\right)^2 = 12$.
• Step 4: Solve $\left(x+7\right)^2 = 12$ taking the square root of both sides, resulting in $x + 7 = \pm\sqrt{12}$.
• Step 5: Simplify and solve for $x$ to find: $x = -7 \pm 2\sqrt{3}$.

Step 6: Now, determine the positive and negative domains:

• The roots are $x = -7 + 2\sqrt{3}$ and $x = -7 - 2\sqrt{3}$.
• Since the parabola opens downward, the function is positive between the roots $-7 - 2\sqrt{3}$ and $-7 + 2\sqrt{3}$, where $x \gt 0$.
• The function is negative for $x \gt -7 + 2\sqrt{3}$ and $x \lt -7 - 2\sqrt{3}$.

Therefore, the solution to the problem is:
Positive domain: $-7 - 2\sqrt{3} < x < -7 + 2\sqrt{3}$
Negative domain: $x > -7 + 2\sqrt{3}$ or $x < -7 - 2\sqrt{3}$.

As outlined between the choice options, the correct answer is represented under choice 4:

$x > -7+2\sqrt{3}$ or $x < 0 : x < -7-2\sqrt{3}$

$x > 0 : -7-2\sqrt{3} < x < -7+2\sqrt{3}$