Find the Domain of y=-2x²+22⅔: Positive and Negative Analysis

To solve this problem, we'll follow these steps:

• Step 1: Convert mixed number $22\frac{2}{9}$ to an improper fraction.
• Step 2: Use the quadratic formula to find the roots of the function.
• Step 3: Determine the intervals where $y > 0$ and $y < 0$.

Now, let's work through each step:

Step 1: Convert $22\frac{2}{9}$ to an improper fraction:

$22\frac{2}{9}$ is $\frac{200}{9} = \frac{200}{9}$.

Step 2: Use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to find the roots.

Here, $a = -2$, $b = 0$, and $c = \frac{200}{9}$.

Calculate the discriminant:

$b^2 - 4ac = 0 - 4(-2)\left(\frac{200}{9}\right) = \frac{1600}{9}$.

The roots become:

$x = \frac{0 \pm \sqrt{\frac{1600}{9}}}{-4} = \frac{\pm \frac{40}{3}}{-4} = \frac{\pm 10}{3}$.

Thus, the roots are $x_1 = -\frac{10}{3}$ and $x_2 = \frac{10}{3}$.

Step 3: Examine where the quadratic is positive:

- The function is shaped as a downward-opening parabola. Its positive zone (above x-axis) will be between the roots.

- So, the positive domain is: $-\frac{10}{3} < x < \frac{10}{3}$.

- Outside these roots, the function is negative:

- The negative domain corresponds to intervals $x < -\frac{10}{3}$ or $x > \frac{10}{3}$.

Therefore, the solution to the problem is:

$x > 3\frac{1}{3}$ or $x < 0 : x < -3\frac{1}{3}$

and

$x > 0 : -3\frac{1}{3} < x < 3\frac{1}{3}$.