1. **Stating the problem:** We have a quadratic function $y = ax^2 + bx + c$ with conditions $x(0) = 2$ and $y(0) = -3$. We want to find the value of $\frac{a \cdot c}{b}$. 2. **Interpreting the conditions:** Usually, $x(0)$ means the value of $x$ at some parameter 0, but here it seems to be a value of $x$ at 0, which is unusual. More likely, the problem means $x=0$ and $y=2$, and $y=0$ and $y=-3$ at some points. But since $x(0)=2$ is given, it might be a typo or misinterpretation. Assuming $x=0$ and $y=2$ means $y(0) = 2$, but the problem states $y(0) = -3$. So we take $x=0$ and $y=-3$. 3. **Using $y(0) = -3$:** Substitute $x=0$ into $y = ax^2 + bx + c$: $$y(0) = a \cdot 0^2 + b \cdot 0 + c = c = -3$$ So, $c = -3$. 4. **Using $x(0) = 2$:** This is ambiguous, but if it means the function value at $x=0$ is 2, it contradicts $y(0) = -3$. If it means $x=0$ corresponds to $y=2$, that conflicts. Alternatively, if $x(0) = 2$ means the root of the function at $x=0$ is 2, or $x=0$ maps to 2, it's unclear. Since $x(0)$ is not standard notation, we assume it means the function value at $x=0$ is 2, but that conflicts with $y(0) = -3$. So we consider $x=0$ and $y=-3$ only. 5. **Without more info, we cannot find $a$ or $b$ directly.** The problem asks for $\frac{a \cdot c}{b}$. 6. **Since $c = -3$, the expression becomes $\frac{a \cdot (-3)}{b} = -3 \frac{a}{b}$.** 7. **Without additional conditions or values for $a$ and $b$, $\frac{a \cdot c}{b}$ cannot be determined uniquely.** **Final answer:** Cannot determine $\frac{a \cdot c}{b}$ with given information.