Solving y = (1/2)x² + 4⅗x: Finding Where Function is Positive

To solve the problem of finding for which values of $x$ the function $y = \frac{1}{2}x^2 + 4\frac{3}{5}x$ is positive, follow these steps:

• Step 1: Identify the coefficients of the quadratic function. Here $a = \frac{1}{2}$, $b = 4\frac{3}{5} = \frac{23}{5}$, and $c = 0$.
• Step 2: Apply the quadratic formula to find the roots. Since $c = 0$, the formula simplifies, and we focus on:
• $x = \frac{-\frac{23}{5} \pm \sqrt{\left(\frac{23}{5}\right)^2 - 4 \cdot \frac{1}{2} \cdot 0}}{1}$
• Calculating the discriminant: $\left(\frac{23}{5}\right)^2 = \frac{529}{25}$. Therefore, roots occur at:
• $x = 0 \quad \text{and} \quad x = -\frac{23}{5} = -4.6$
• Step 3: Analyze the sign of $f(x)$ around the roots. A parabola opens upwards (since $a = \frac{1}{2} > 0$), so $f(x)$ is positive when beyond these roots.
• The intervals to check are $x < -4.6$ and $x > 0$ where $f(x) > 0$.

Therefore, the solution to the problem is $x > 0$ or $x < -4.6$, which translates to the fractional values $x > 0$ or $x < -9\frac{1}{5}$, matching choice 2.