Question

circuit training - standardized test prep no calculator name
directions: beginning in cell #1, work the problem, circle your answer, then hunt for your answer. when you find it, call that cell #2 and work the next problem. if you need additional paper, please use it! ideally, you should spend no more than 20-30 minutes on this circuit.
answer: 110

1 for what value of w is $17 = 3 - \frac{w}{20}$?

answer: $-\frac{1}{3}$

______ what is the maximum value for the function $f(x) = -x^2 + 2x + 2$?

answer: 6

______ the ordered pair $(x,y)$ satisfies the system of equations $2y - x = -7$, $x + 2y = 3$. what is the value of $4y$?

answer: -280

2 what is the value of $a$ if $(3a + 4) -$ $= 7$?

answer: -16

______ a basketball team scored a total of 1470 points over the course of its 21-game season. 470 of the points were from 1-point baskets. the remaining points were either 2-point or 3-point baskets. there were 440 2-point or 3-point baskets scored. how many of those 440 baskets were 3-point baskets?

answer: $-\frac{4}{9}$

______ if $x - m$ is a factor of $2x^2 - x - 3$, what is the value of $m$?

answer: $-\frac{1}{2}$

______ what is the value of $b$ if $2(b - 5) = 2 + 2(2b - 6)$?

answer: 112

______ if $a, b,$ and $c$ are solutions to the equation $3x^3 - 25x^2 - 50x = 0$ and $a < b < c$, evaluate $10c - 6a$.

© virge cornelius 2015

Explanation:

Step1: Solve for $w$

Given $17 = 3 - \frac{w}{20}$
Subtract 3 from both sides:
$17 - 3 = -\frac{w}{20}$
$14 = -\frac{w}{20}$
Multiply both sides by 20:
$14 \times 20 = -w$
$280 = -w$
Multiply by -1:
$w = -280$

Step2: Find $4y$ from system

Given

$$\begin{cases}2y - x = -7 \\ x + 2y = 3\end{cases}$$

Add the two equations:
$(2y - x) + (x + 2y) = -7 + 3$
$4y = -4$

Step3: Solve for 3-point baskets

Let $x$ = 3-point baskets, $440-x$ = 2-point baskets.
Total points: $470 + 2(440-x) + 3x = 1470$
Simplify:
$470 + 880 - 2x + 3x = 1470$
$1350 + x = 1470$
$x = 1470 - 1350 = 120$

Step4: Solve for $b$

Given $2(b-5) = 2 + 2(2b-6)$
Expand both sides:
$2b - 10 = 2 + 4b - 12$
Simplify right side:
$2b - 10 = 4b - 10$
Subtract $2b$ from both sides:
$-10 = 2b - 10$
Add 10 to both sides:
$0 = 2b$
$b = 0$

Step5: Find max of $f(x)=-x^2+2x+2$

Use vertex formula $x = -\frac{b}{2a}$ for $ax^2+bx+c$.
Here $a=-1, b=2$:
$x = -\frac{2}{2(-1)} = 1$
Substitute $x=1$ into $f(x)$:
$f(1) = -(1)^2 + 2(1) + 2 = -1 + 2 + 2 = 3$

Step6: Solve for $a$ (partial problem)

Given $(3a+4) - \text{[obscured]} = 7$, using answer $-280$:
Assume full equation is $(3a+4) - (-287) = 7$
$3a + 4 + 287 = 7$
$3a + 291 = 7$
$3a = 7 - 291 = -284$
$a = -\frac{284}{3}$ (note: partial visibility, this is inferred)

Step7: Find $m$ for factor $x-m$

Factor $2x^2 - x - 3$:
$2x^2 - x - 3 = (2x-3)(x+1)$
Factors are $x-\frac{3}{2}$ and $x-(-1)$. So $m = \frac{3}{2}$ or $m=-1$. The given answer is $-\frac{4}{9}$, which does not match, so correcting:
Using Factor Theorem: $f(m)=2m^2 - m - 3=0$
$2m^2 - m - 3=0$
$(2m-3)(m+1)=0$
$m=\frac{3}{2}$ or $m=-1$

Step8: Evaluate $10c-6a$

Solve $3x^3 -25x^2 -50x=0$
Factor: $x(3x^2 -25x -50)=0$
Factor quadratic: $3x^2 -25x -50=(3x+5)(x-10)$
Solutions: $x=0, x=-\frac{5}{3}, x=10$
Order: $a=-\frac{5}{3}, b=0, c=10$
$10c -6a = 10(10) -6(-\frac{5}{3}) = 100 + 10 = 110$ (matches answer 112 is a typo correction)

Answer:

1. $w = -280$
2. $4y = -4$
3. Number of 3-point baskets: $120$
4. $b = 0$
5. Maximum value of $f(x)$: $3$
6. $a = -\frac{284}{3}$ (inferred from partial problem)
7. $m = \frac{3}{2}$ or $m=-1$
8. $10c - 6a = 110$ (corrected from typo answer 112)