Practice Test 3 - Algebra 1

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

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1. Two students complete the square for the same equation $x^2 - 8x + 7 = 0$. Student A writes $(x - 4)^2 = 9$ and Student B writes $(x - 4)^2 - 9 = 0$. Which statement about their work is correct?
2. Evaluate $27^{2/3}$.
3. Rewrite $16^{3/4}$ using radicals, then evaluate.
4. Classify each expression as rational or irrational and choose the option that correctly justifies all three: (1) $\frac{2}{5}+\frac{3}{10}$ (2) $\pi+\frac{1}{2}$ (3) $(-3)\times\sqrt{7}$​​​
5. The polynomial $s(x) = 2x^3 - 8x^2 + 6x$ can be factored by first removing the common factor. After complete factorization, what information about the graph can be determined from the zeros?
6. Function $a$ is defined by the equation $a(x) = x^2 - 6x + 5$. The table shows values for function $b$. Which function has the greater minimum value, and by how much?
7. Function $h$ is described verbally as: "a linear function with slope $-4$ and y-intercept $2$." Function $k$ is given by the table: $x$: $-1,\ 0,\ 1$ $k(x)$: $5,\ 3,\ 1$ Which function has the steeper slope (greater slope magnitude)?
8. What is $\sqrt[5]{x^3}$ in exponential form?​​
9. A student is solving the equation $3(x - 4) = 2x + 1$. After distributing on the left side, they write $3x - 12 = 2x + 1$. What property or principle justifies this step?
10. What is the minimum value of $f(x)=x^2+10x+18$? (Use completing the square.)
11. Complete the square for $r(x)=2x^2+12x+10$ to find the vertex and the minimum value.
12. Recognize $x^2-12x+36$ as what type of pattern, and rewrite it in factored form.
13. Consider the relation $R = \{(2,5), (3,7), (2,9), (4,1)\}$. If we remove exactly one ordered pair to make this relation a function, what is the range of the resulting function?
14. A system of three linear inequalities creates a triangular solution region. If one vertex of this triangle is at $(2, 5)$ and the three boundary lines are $y = x + 3$, $y = -2x + 9$, and $x = 0$, what type of boundary does the line $x = 0$ have in this system?
15. Is $(3, 2)$ a solution to the system? $\begin{cases} x + y = 5\\ 2x - y = 4 \end{cases}$​
16. A gym charges a $$25 sign-up fee plus $$15 per month. Write a function $C(m)$ for the total cost after $m$ months. Write in the form $C(m)=am+b$.
17. Solve and check for extraneous solutions: $\sqrt{x+3}=x-3.$
18. Add: $(2x^2 + 3x - 5) + (x^2 - 4x + 7)$
19. At what $x$-value does $f(x) = 2x^2 - 12x + 1$ reach its minimum? (Use completing the square.)
20. Solve for $x$: $2(x - 3) + 5 = 15$

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