Find the Domains: Analyzing y = -x² + 5x + 14 Quadratic Function

We will analyze the function $y = -x^2 + 5x + 14$ to find where it is positive and negative. Let's begin by finding its roots:

• Step 1: Calculate the discriminant for $-x^2 + 5x + 14 = 0$:
  The discriminant $\Delta = b^2 - 4ac = 5^2 - 4(-1)(14) = 25 + 56 = 81$.
• Step 2: Find the roots using the quadratic formula $x = \frac{-b \pm \sqrt{\Delta}}{2a}$:
  Here $a = -1$, $b = 5$, and $\Delta = 81$, so $x = \frac{-5 \pm \sqrt{81}}{-2}$.
  $x = \frac{-5 \pm 9}{-2}$, giving roots $x_1 = -\frac{14}{-2} = 7$ and $x_2 = 2$.
• Step 3: Identify the intervals on the number line divided by these roots: $(-\infty, 2)$, $(2, 7)$, and $(7, \infty)$.
• Step 4: Test each interval to determine if the function is positive or negative:
• For $x < 2$: Choose $x = 0$: $y = 14$, $y > 0$.
• For $2 < x < 7$: Choose $x = 5$: $y = -5^2 + 5 \times 5 + 14 = -25 + 25 + 14 = 14$, $y > 0$.
• For $x > 7$: Choose $x = 10$: $y = -10^2 + 5 \times 10 + 14 = -100 + 50 + 14 = -36$, $y < 0$.
• Summary of intervals:
  $y > 0$ for $-2 < x < 7$.
  $y < 0$ for $x < -2$ or $x > 7$.

Therefore, the positive domain of the function is $(-2, 7)$ and the negative domain is $(-\infty, -2)$ or $(7, \infty)$.

The correct choice matching our solution is Choice 2:
$x > 0 : -2 < x < 7$
$x > 7$ or $x < 0 : x < -2$