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1. Cube root plus square root equation ●●○○○ 20 ptsSolve the equation $\sqrt[3]{x} + \sqrt{x+12} = 0$ for $x$.
2. Value of f(2) for a quadratic ●●●○○ 50 ptsLet $a$ and $b$ be real numbers. The quadratic function $f(x)=x^{2}+ax+b$ satisfies $f(f(0))=f(f(1))=0$ and $f(0)\neq f(1)$. Find $f(2)$.
3. Minimum roots of g times g' ●●●○○ 50 ptsLet $f$ be a function defined for all $x \in \mathbb{R}$ that is discontinuous at exactly $3$ values of $x$. Let $g$ be a differentiable function such that $f(x)g(x)$ is continuous for all $x \in \mathbb{R}$. Find the minimum number of distinct real roots of the equation $g(x)g'(x)=0$.
4. Maximum number of quadratic equations on a board ●●●○○ 50 ptsWhat is the maximum number of different monic quadratic equations that can be written on a board, given that any two of them have a common root, but no four of them have a root common to all?