Solve y = 1/4x² - 3.5x: Finding Negative Function Values

To determine when $f(x) = \frac{1}{4}x^2 - 3\frac{1}{2}x < 0$, follow these steps:

Step 1: Find the roots of the quadratic equation.

The function can be rewritten as $y = \frac{1}{4}x^2 - \frac{7}{2}x$. Set this equal to zero to find the roots:

$\frac{1}{4}x^2 - \frac{7}{2}x = 0$

Factor out $x$: $x\left(\frac{1}{4}x - \frac{7}{2}\right) = 0$

So, $x = 0$ or $\frac{1}{4}x = \frac{7}{2}$. Solve the second equation:

$x = \frac{7}{2} \times 4 = 14$

Step 2: Analyze the intervals around the roots.

The roots are $x = 0$ and $x = 14$. These divide the number line into three intervals: $x < 0$, $0 < x < 14$, and $x > 14$.

Step 3: Perform a sign test in each interval.

• Test for $x < 0$: Choose $x = -1$. The value of the function $f(-1) = \frac{1}{4}(-1)^2 - \frac{7}{2}(-1) = \frac{1}{4} + \frac{7}{2} > 0$.
• Test for $0 < x < 14$: Choose $x = 7$. The value of the function $f(7) = \frac{1}{4}(7)^2 - \frac{7}{2}(7) = \frac{49}{4} - \frac{49}{2} = \frac{49}{4} - \frac{98}{4} = -\frac{49}{4} < 0$.
• Test for $x > 14$: Choose $x = 15$. The value of the function $f(15) = \frac{1}{4}(15)^2 - \frac{7}{2}(15) = \frac{225}{4} - \frac{105}{2} = \frac{225}{4} - \frac{210}{4} = \frac{15}{4} > 0$.

Conclusion: The quadratic $\frac{1}{4}x^2 - \frac{7}{2}x$ is less than zero for $0 < x < 14$.

Therefore, the solution to the problem is $0 < x < 14$.