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1. Coefficients of a re-written quadratic ●○○○○ 10 ptsThe expression $2(x-3)^{2}-12$ can be re-written as $ax^{2}+bx+c$ for some numbers $a, b, c$. What is the value of $10a-b-4c$?
2. Sum of squares after two increments ●●○○○ 20 ptsEach of 100 given numbers was increased by $1$. Then each number was increased by $1$ once more. Given that after the first increase the sum of the squares of the numbers did not change, find by how much the sum of the squares changed after the second increase.
3. Widget output difference on two days ●●○○○ 20 ptsOn Monday, David produces $w$ widgets per hour and works for $t$ hours. On Tuesday, he works for $2$ fewer hours than on Monday, but produces $4$ additional widgets per hour. If $w = 2t$, how many more widgets did David produce on Monday than on Tuesday?
4. Quadratic constraint on a shifted expression ●●○○○ 20 ptsGiven $6x^{2}-4x-3=0$, find the value of $(x-1)^{2}+x\left(x+\frac{2}{3}\right)$.
5. Function value via substitution ●●○○○ 20 ptsGiven the function $f(x-1)=x^{2}-2x$, find the value of $f(\sqrt{2})$.
6. Solve an equation with expanded products ●●○○○ 20 ptsSolve the equation $6(x-1)(x+2)-4(x-3)(x+4)=2(x-5)(x-6)$.