Find the minimum value of the curve 12x2 + 24x + 13. 11. If the roots of ax2 + bx + c = 0 are alpha and beta, show that alpha + beta = -(b)/(a) and alphabeta = (c)/(a).

$$x = -\frac{24}{2(12)}$$ $$x = -\frac{24}{24}$$ $$x = -1$$

$$y = 12(-1)^2 + 24(-1) + 13$$ $$y = 12(1) - 24 + 13$$ $$y = 12 - 24 + 13$$ $$y = -12 + 13$$ $$y = 1$$

The minimum value of the curve is $\boxed{1}$.

11. If the roots of $ax^2 + bx + c = 0$ are $\alpha$ and $\beta$, show that $\alpha + \beta = -\frac{b}{a}$ and $\alpha\beta = \frac{c}{a}$.

The roots of the quadratic equation $ax^2 + bx + c = 0$ are given by:

$$\alpha = \frac{-b + \sqrt{b^2 - 4ac}}{2a}$$ $$\beta = \frac{-b - \sqrt{b^2 - 4ac}}{2a}$$

$$\alpha + \beta = \frac{-b + \sqrt{b^2 - 4ac}}{2a} + \frac{-b - \sqrt{b^2 - 4ac}}{2a}$$ $$\alpha + \beta = \frac{(-b + \sqrt{b^2 - 4ac}) + (-b - \sqrt{b^2 - 4ac})}{2a}$$ $$\alpha + \beta = \frac{-b - b + \sqrt{b^2 - 4ac} - \sqrt{b^2 - 4ac}}{2a}$$ $$\alpha + \beta = \frac{-2b}{2a}$$ $$\alpha + \beta = -\frac{b}{a}$$

$$\alpha\beta = \left(\frac{-b + \sqrt{b^2 - 4ac}}{2a}\right) \left(\frac{-b - \sqrt{b^2 - 4ac}}{2a}\right)$$

This is in the form $(X+Y)(X-Y) = X^2 - Y^2$, where $X = -b$ and $Y = \sqrt{b^2 - 4ac}$.

$$\alpha\beta = \frac{(-b)^2 - (\sqrt{b^2 - 4ac})^2}{(2a)^2}$$ $$\alpha\beta = \frac{b^2 - (b^2 - 4ac)}{4a^2}$$ $$\alpha\beta = \frac{b^2 - b^2 + 4ac}{4a^2}$$ $$\alpha\beta = \frac{4ac}{4a^2}$$ $$\alpha\beta = \frac{c}{a}$$

Thus, we have shown that $\boxed{\alpha + \beta = -\frac{b}{a}}$ and $\boxed{\alpha\beta = \frac{c}{a}}$.

12. Find $b$ if $h(x) = 2x-1$ is a factor of $f(x) = 6x^2 + bx + 2$.

If $h(x)$ is a factor of $f(x)$, then $f(k)=0$ for the root $k$ of $h(x)=0$.
First, find the root of $h(x) = 2x-1$:

$$2x - 1 = 0$$ $$2x = 1$$ $$x = \frac{1}{2}$$

$$6\left(\frac{1}{4}\right) + \frac{b}{2} + 2 = 0$$ $$\frac{6}{4} + \frac{b}{2} + 2 = 0$$ $$\frac{3}{2} + \frac{b}{2} + 2 = 0$$

To eliminate fractions, multiply the entire equation by 2:

$$2\left(\frac{3}{2}\right) + 2\left(\frac{b}{2}\right) + 2(2) = 2(0)$$ $$3 + b + 4 = 0$$ $$b + 7 = 0$$ $$b = -7$$

The value of $b$ is $\boxed{-7}$.