Readiness Check for Algebra 2: Advanced Algebra Solutions

Show all work.

1. Solve.
     $\frac{x–6}{3}=7\left(x+2\right)$
   
   $\begin{array}{rcl}\left(3\right)\frac{x–6}{3}& =& 7\left(x+2\right) \left(3\right)\\ x–6& =& 21\left(x+2\right)\\ x–6& =& 21x+42\\ & & –x–42 –x–42\\ –\frac{48}{20}& =& \frac{20x}{20}\\ x& =& –\frac{12}{5}\end{array}$

2. Write the equation in terms of b.
   $y=a{x}^2+bx+c$
   
   $\begin{array}{rcl}y& =& a{x}^2+bx+c \\ & & –a{x}^2–c –a{x}^2 –c\\ \frac{–a{x}^2–c+y}{x}& =& \frac{bx}{x}\\ b& =& \frac{–a{x}^2–c+y}{x}\end{array}$

3. Simplify.
   $\frac{11a^5{b}^{–3}}{c^{–8}}{\left(\frac{b^2}{a{c}^4}\right)}^3$
   
   $$\begin{gathered} \frac{11a^5{b}^{–3}}{c^{–8}}·\frac{b^6}{a^3{c}^{12}} \\ \frac{11a^{5–3}{b}^{–3+6}}{c^{–8+12}}=\frac{11a^2{b}^3}{c^4} \end{gathered}$$

4. Write the answer in simplified radical form.
   $\sqrt{9x^3{y}^4}$
   
   $$\begin{gathered} \sqrt{3^2 x^3 y^4} \\ {3}^{\frac{2}{2}} x^{\frac{3}{2}} y^{\frac{4}{2}}=3^1 x^{1\frac{1}{2} }{y}^2 \\ 3x{y}^2\sqrt{x} \end{gathered}$$

7. Simplify.
   $\left(2x^2–7x+8\right)–3\left(x^2–4x\right)$
   $$\begin{gathered} 2x^2–7x+8–3x^2+12x \\ –x^2+5x+8 \end{gathered}$$

8. Classify the polynomial.
   $15x^3+x–7$
   cubic trinomial 

9. Solve the equation for b. 
   $A=bh+2jk+bk$
   $\begin{array}{rcl}A& =& bh+2jk+bk \\ & & –2jk –2jk\\ A–2jk& =& bh+bk\\ \frac{A–2jk}{h+k}& =& \frac{b\left(h+k\right)}{h+k}\\ b& =& \frac{A–2jk}{h+k}\end{array}$

10. Solve.
    $\frac{1}{5}\left(2x–3\right)=\frac{3}{2}x–1$
    $\begin{array}{rcl}\mathrm{LCD}\left(5, 2\right)& =& 10\\ \left(10\right)\frac{1}{5}\left(2x–3\right)& =& \left(\frac{3}{2}x–1 \right)\left(10\right)\\ 2\left(2x–3\right)& =& 15x–10\\ 4x–6& =& 15x–10\\ & & –4x+10 –4x+10\\ \frac{4}{11}& =& \frac{11x}{11}\\ x& =& \frac{4}{11}\end{array}$

11. Factor completely.
    $16x^3+48x^2–x–3$
    $$\begin{gathered} \left(16x^3+48x^2\right)+\left(–x–3\right) \\ 16x^2\left(x+3\right)–1\left(x+3\right) \\ \left(x+3\right)\left(16x^2–1\right) \\ \left(x+3\right)\left(4x–1\right)\left(4x+1\right) \end{gathered}$$

12. Simplify.
    ${\left(3x^3{y}^{2}z\right)}^4 \left(\frac{y^5{z}^{–6}}{12x^{25}{z}^5}\right)$
    $$\begin{gathered} {\left(3x^3{y}^{2}z\right)}^4 \left(\frac{y^5}{12x^{25}{z}^{5–\left(–6\right)}}\right) \\ {3}^4 x^{12} y^8 z^4 \left(\frac{y^5}{12x^{25}{z}^{11}}\right) \\ \frac{81y^{8+5}}{12x^{25–12} z^{11–4}}=\frac{27y^{13}}{4x^{13} z^7} \end{gathered}$$

13. Explain why the relation does or does not represent a function.
    {(–5, 1), (–3, –3), (1, –3), (3, –5)}
    This is a function because the domain values do not repeat.

14. Describe the transformation of the quadratic equation from the parent function.
    $y=–{\left(x–2\right)}^2+5$
    a = –1, h = 2, k = 5
    The graph reflects, then shifts right 2 spaces and up 5 spaces from the parent graph.

15. Solve the system.
    $$\begin{gathered} y=\frac{4}{3}x–20 \\ 2x+9y=72 \end{gathered}$$
    
      $\begin{array}{rcl}2x+9\left(\frac{4}{3}x–20\right)& =& 72 \\ 2x+12x–180& =& 72\\ 14x–180& =& 72\\ & & +180 +180\\ \frac{14x}{14}& =& \frac{252}{14} y=\frac{4}{3}\left(18\right)–20\\ x& =& 18 y=4\\ & & \left(18, 4\right)\end{array}$

17. Factor completely.
    $10x^{2}y–54x{y}^2–36y^3$
     $$\begin{gathered} 2y\left(5x^2–27xy–18y^2\right) \\ 2y\left(5x+3y\right)\left(x–6y\right) \end{gathered}$$

18. Write the answer in simplified radical form.
    $\sqrt[3]{8a^8{b}^5{c}^{15}}$
      $$\begin{gathered} \sqrt[3]{2^3·3 a^8 b^5 c^{15}} \\ {2}^{\frac{3}{3}}{3}^{\frac{1}{3}}{a}^{\frac{8}{3}}{b}^{\frac{5}{3}}{c}^{\frac{15}{3}}=2^1·3^{\frac{1}{3} }{a}^{2\frac{2}{3} }{b}^{1\frac{2}{3}} c^5 \\ 2a^{2}b{c}^5 \sqrt[3]{3a^2{b}^2} \end{gathered}$$

19. Describe the transformation of the quadratic equation from the parent function. 
    $y=2{\left(x+8\right)}^2–1$
    $a = 2, h = –8, k = –1$
    The graph dilates by a factor of 2, so the graph will be narrower than the parent graph. It also shifts left 8 spaces and down 1 space from the parent graph.

20. Solve the system of equations.
    $$\begin{gathered} 5x+7y=14 \\ 3x–y=24 \end{gathered}$$ 
    $$\begin{gathered} 5x+7y = 14 \\ \left(3x–y=24\right)\left(7\right)\Rightarrow \underline{+21x–7y= 168} \\ \frac{26x}{26}=\frac{182}{26} \\ x=7 \end{gathered}$$
    $$\begin{gathered} 5\left(7\right)+7y=14 \\ 35+7y=14 \\ –35 –35 \\ \frac{ 7y}{7}=–\frac{21}{7} \\ y=–3 \left(7, –3\right) \end{gathered}$$