Exploring Positive and Negative Domains: y = x² + 2x + 4.2

Find the positive and negative domains of the following function:

$y=x^2+2x+4\frac{1}{5}$

To find the positive and negative domains of the function $y = x^2 + 2x + 4.2$, we need to consider the graph of this function and its roots.

First, let's compute the discriminant of the quadratic $y = x^2 + 2x + 4.2$. The discriminant $\Delta$ is given by $b^2 - 4ac$.

Here, $a = 1$, $b = 2$, and $c = 4.2$.

Calculating, we have:

$\Delta = 2^2 - 4 \cdot 1 \cdot 4.2 = 4 - 16.8 = -12.8$.

Since the discriminant is negative, there are no real roots. This means the parabola does not intersect the x-axis.

Next, because $a = 1$ is positive, the parabola opens upwards.

Hence, the entire parabola lies above the x-axis, indicating that the function $y = x^2 + 2x + 4.2$ is positive for all real $x$.

Thus, there is no negative domain for this quadratic since it doesn't dip below the x-axis at any point.

Therefore, the positive and negative domains are:

$x > 0 :$ for all $x$

$x < 0 :$ none