Determine the Positive and Negative Domains of y = 25x² + 20x + 4

Find the positive and negative domains of the function below:

$y=25x^2+20x+4$

To determine the positive and negative domains of $y = 25x^2 + 20x + 4$, we follow these steps:

• Step 1: Calculate the discriminant $\Delta = b^2 - 4ac = 20^2 - 4 \cdot 25 \cdot 4 = 400 - 400 = 0$.
• Step 2: Since the discriminant is zero, the quadratic has one distinct real root, given by $x = \frac{-b}{2a} = \frac{-20}{2 \cdot 25} = -\frac{2}{5}$.
• Step 3: Analyze the sign of $y = 25x^2 + 20x + 4$.

Since the discriminant is zero, the quadratic equation reaches zero only at $x = -\frac{2}{5}$, and is symmetrical around this point. Therefore:

• For $x > 0$, $y$ is positive except where $x = -\frac{2}{5}$.
• For $x < 0$, there are no points where the graph is negative since the parabola opens upwards and does not cross below the x-axis after this root.

Thus, interpreting the domains:

The positive domain for $x > 0$ is all $x$ except $x = -\frac{2}{5}$. For $x < 0$, there is no negative domain because the graph does not descend below the x-axis.

Therefore, the solution to the problem is:

$x > 0 :x\ne\frac{2}{5}$

$x < 0 :$ none