Explore the Quadratic: Find Positive and Negative Domains of y = -1/2x² + x - 1

Find the positive and negative domains of the following function:

$y=-\frac{1}{2}x^2+x-1$

To solve this problem, we'll determine where the quadratic function $y = -\frac{1}{2}x^2 + x - 1$ is positive and negative.

Step 1: Find the roots of the quadratic equation.
We'll use the quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = -\frac{1}{2}$, $b = 1$, and $c = -1$.

Calculate the discriminant $b^2 - 4ac = 1^2 - 4(-\frac{1}{2})(-1) = 1 - 2 = -1$.
Notice that the discriminant is negative, meaning the quadratic equation has no real roots.

Step 2: Determine the sign of the quadratic function.
Since there are no real roots, the quadratic does not intersect the x-axis. Since $a = -\frac{1}{2}$, which is negative, the parabola opens downwards. Without real roots, it means it is always negative for all values of $x$.

Conclusion: The function is negative for all $x$.

Therefore, the positive and negative domains of the function are:

$x < 0 :$ for all $x$

$x > 0 :$ none