Practice Test 1 - Algebra 1

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

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1. Solve $x^2 + 2x + 2 = 0$. Express complex solutions in $a \pm bi$ form.
2. What value of $x$ makes the equation true? $\frac{x}{3} + 2 = 7$
3. Is the point $(-1, 2)$ on the graph of $y = |x| + 1$? Use substitution to decide.
4. Subtract: $(6x^2 + 2x - 9) - (4x^2 - 5x + 1)$
5. The sequence $1, 1, 2, 3, 5, \dots$ is defined by $f(1)=1$, $f(2)=1$, and $f(n)=f(n-1)+f(n-2)$ for $n\ge 3$ (with domain $n=1,2,3,\dots$). What is $f(6)$?
6. A video game score starts at 80 points and increases by 25 points each level completed. Write an explicit formula for the score $S(n)$ after completing $n$ levels, where $n=0$ means no levels completed yet.
7. How many real solutions does the system have? (Equivalently: how many intersection points are there between the line and the parabola?) $\begin{cases} y = x^2 + 2x + 5 \\ y = -x + 1 \end{cases}$
8. Derive the quadratic formula by completing the square on the general equation $ax^2 + bx + c = 0$ (with $a\ne 0$). Which expression for $x$ results?
9. Which situation involves a constant percent rate of change (exponential) rather than a constant additive change (linear)?
10. A cell phone plan charges a $25 monthly fee plus $0.10 per text message. After analyzing usage patterns, the company offers a new plan with a $35 monthly fee and $0.05 per text message. For what range of monthly text messages would the new plan result in a constant rate of savings compared to the original plan?
11. Which explicit formula represents the sequence $2,\ 6,\ 18,\ 54,\ \dots$?
12. In the formula for pressure, $P=\dfrac{F}{A}$, force $F$ is measured in newtons (N) and area $A$ is measured in square meters ($\text{m}^2$). What are the units of $P$?
13. For the explicit formula $a_n = 5\cdot 2^{n-1}$, translate it to a recursive definition.
14. Based on the table shown, what is the average rate of change of $f(x)$ from $x = -1$ to $x = 5$?
15. A teacher defines $b^{1/n}=\sqrt[n]{b}$ (for $b>0$) and then defines $b^{m/n}=\left(b^{1/n}\right)^m$. Which argument shows this is not arbitrary but forced by wanting exponent rules to extend from integers to rationals?
16. Solve $x^2 + 8x + 20 = 0$ and write solutions in the form $a \pm bi$. (Use $i^2 = -1$.)
17. What is the simplified form of the rational expression (a rational expression remains rational after simplification, showing closure under equivalent transformations)? $\frac{x^2-5x+6}{x^2-9}$
18. Simplify the rational expression by factoring and canceling common factors: $\frac{x^2-9}{x+3}$
19. If $h(x) = x - 3$, what is the value of $h(0)$?
20. What are the main parts of the expression $\sqrt{(x-4)^2+9}$? View $(x-4)$ as a single unit.

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