Determine Positive and Negative Domains of the Function: y = x² - 22x + 121

To determine the positive and negative domains of the quadratic function $y = x^2 - 22x + 121$, we start by finding its roots. The roots will help us identify intervals where the function is positive or negative.

First, apply the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

The given quadratic equation is $x^2 - 22x + 121$. Here, $a = 1$, $b = -22$, and $c = 121$. Calculate the discriminant:

$\Delta = b^2 - 4ac = (-22)^2 - 4 \times 1 \times 121 = 484 - 484 = 0$

The discriminant is zero, indicating a repeated root. Applying the quadratic formula gives:

$x = \frac{-(-22) \pm \sqrt{0}}{2 \times 1} = \frac{22 \pm 0}{2} = 11$

This means there is only one root, $x = 11$. A quadratic with a double root at $x = 11$ indicates the function touches the x-axis at $x = 11$ and opens upwards (since $a > 0$). This implies that the function is non-negative for all $x$. Therefore, the function does not have a negative domain.

As the quadratic is positive for $x \ne 11$, the positive domain is all $x$ except when $x = 11$.

Therefore, the positive domain is $x > 0 : x \ne 11$.

The negative domain, where the function would take negative values, is nonexistent as the parabola never crosses beneath the x-axis.

The solution to the problem is:

$x > 0 : x \ne 11$

$x < 0 :$ none