Practice Test 38 - Algebra 1

20 questions. Answer them in any order, then submit to see your score and every explanation.

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1. Solve using the quadratic formula: $x^2+6x+25=0$. The discriminant is negative, so the solutions are complex conjugates. Use $i^2=-1$ and write solutions in $a+bi$ form.​
2. The sequence $7, 5, 3, 1, \dots$ is a function whose input is $n = 1,2,3,\dots$ and output is $a_n$. Which recursive rule generates this sequence?
3. A bacteria culture starts with 1000 cells and increases by 20% each hour. Which statement correctly identifies the growth factor $b$ and the percent rate $r$ per hour in the exponential form $N(t)=a\cdot b^t$?
4. A water tank is being filled at a rate of 15 gallons per minute for the first 20 minutes, then at 8 gallons per minute for the next 30 minutes. Which statement best describes the rate of change of water volume with respect to time?
5. If $y = x^{\frac{3}{4}}$, then $y^{\frac{4}{3}}$ equals which expression?
6. The quadratic equation $x^2 - 4x + k = 0$ is transformed by completing the square to $(x - 2)^2 = 7$. What is the value of $k$, and what are the solutions to the original equation?
7. An item's value is modeled by $V(t)=1200(1+r)^t$. If the value decreases by 15% each year, which value of $r$ should be used?
8. A movie theater has 200 seats. The function $R(n)=12n$ gives the revenue (in dollars) from selling $n$ tickets, where $n$ is the number of tickets sold. What is an appropriate realistic domain for $R$?​
9. A sequence is defined by the explicit formula $a_n=12-2n$ for $n=1,2,3,\dots$. What is $a_4$?
10. Use the zeros and degree to sketch a rough graph of $P(x)=(x+2)^2(x-3)$. Which statement is correct about x-intercepts and end behavior?
11. Identify the error in the student work shown for solving $4x-9=3$. Student work: 1) $4x-9=3$ 2) $4x=12$ 3) $x=12-4$ 4) $x=8$ Which statement best identifies the first incorrect step and why it is invalid?
12. A bakery's daily profit $P$ depends on the number of specialty cakes $c$ sold according to $P = 15c - 45$. When graphing this relationship, what do the intercepts represent in this business context?
13. What is the common ratio of the geometric sequence $10,\ 5,\ 2.5,\ 1.25,\ \dots$?
14. Factor $P(x)=x^3-4x$ and use the zeros to choose the correct list of x-intercepts for the graph of $y=P(x)$.
15. A small theater must follow these constraints for ticket sales: $x$ = adult tickets, $y$ = student tickets. $\begin{cases} x + y \le 80 \\ x \ge 10 \\ y \ge 15 \end{cases}$ Graph the feasible region. Which description matches the solution set?
16. Solve the system: $\begin{cases}2x+y=8\\x-y=1\end{cases}$ What is $(x,y)$?
17. An investment account is modeled by $A(t)=1000(1+r)^t$, where $t$ is in years. If the account grows by 8% per year, which function correctly models the balance?
18. Consider the polynomial expression $(x^2 + kx + 3)(2x - 1)$ where $k$ is a constant. If the coefficient of the $x^2$ term in the expanded form is $7$, what is the value of $k$?
19. For the function $y=4x+1$, find both intercepts of the line (as points) to show on the coordinate plane.
20. Find all points $(x,y)$ satisfying both equations (the intersection points): $\begin{cases} y = x - 2 \\ y = x^2 - 4 \end{cases}$

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