Answer Key 7.8

[latexpage]

1. \(\begin{array}{rr}
   \\ \\
   \begin{array}{rrrrr}
   k&-&7&=&0 \\
   &+&7&&+7 \\
   \midrule
   &&k&=&7
   \end{array}
   &\hspace{0.25in}
   \begin{array}{rrrrr}
   k&+&2&=&0 \\
   &-&2&&-2 \\
   \midrule
   &&k&=&-2
   \end{array}
   \end{array}\)
2. \(\begin{array}{rr}
   \\ \\
   \begin{array}{rrrrr}
   a&+&4&=&0 \\
   &-&4&&-4 \\
   \midrule
   &&a&=&-4
   \end{array}
   &\hspace{0.25in}
   \begin{array}{rrrrr}
   a&-&3&=&0 \\
   &+&3&&+3 \\
   \midrule
   &&a&=&3
   \end{array}
   \end{array}\)
3. \(\begin{array}{rr}
   \\ \\
   \begin{array}{rrrrr}
   x&-&1&=&0 \\
   &+&1&&+1 \\
   \midrule
   &&x&=&1
   \end{array}
   &\hspace{0.25in}
   \begin{array}{rrrrr}
   x&+&4&=&0 \\
   &-&4&&-4 \\
   \midrule
   &&x&=&-4
   \end{array}
   \end{array}\)
4. \(\begin{array}{rr}
   \\ \\
   \begin{array}{rrrrr}
   \\ \\ \\
   2x&+&5&=&0 \\
   &-&5&&-5 \\
   \midrule
   &&\dfrac{2x}{2}&=&\dfrac{-5}{2} \\ \\
   &&x&=&-\dfrac{5}{2}
   \end{array}
   &\hspace{0.25in}
   \begin{array}{rrrrr}
   x&-&7&=&0 \\
   &+&7&&+7 \\
   \midrule
   &&x&=&7
   \end{array}
   \end{array}\)
5. \(\begin{array}{rrr}
   \\ \\ \\ \\
   6(x^2-25)&=&0 \\
   6(x-5)(x+5)&=&0 \\ \\
   x&=&5 \\
   x&=&-5
   \end{array}\)
6. \(\begin{array}{rrr}
   \\ \\ \\
   (p+8)(p-4)&=&0 \\ \\
   p&=&-8 \\
   p&=&4
   \end{array}\)
7. \(\begin{array}{rrr}
   \\ \\ \\ \\
   2(n^2+5n-14)&=&0 \\
   2(n+7)(n-2)&=&0 \\ \\
   n&=&-7 \\
   n&=&2
   \end{array}\)
8. \(\begin{array}{rrr}
   \\ \\ \\
   (m-6)(m+5)&=&0 \\ \\
   m&=&6 \\
   m&=&-5
   \end{array}\)
9. \(\begin{array}{rrr}
   \\ \\ \\ \\
   (x+3)(7x+5)&=&0 \\ \\
   x&=&-3 \\
   x&=&-\dfrac{5}{7}
   \end{array}\)
10. \(\begin{array}{rrr}
    \\ \\ \\ \\
    (2b+1)(b-2)&=&0 \\ \\
    b&=&-\dfrac{1}{2} \\ \\
    b&=&2
    \end{array}\)
11. \(\begin{array}{rrrrrrr}
    \\ \\ \\ \\ \\ \\
    x^2&-&4x&-&8&=&-8 \\
    &&&+&8&&+8 \\
    \midrule
    &&x^2&-&4x&=&0 \\
    &&x(x&-&4)&=&0 \\ \\
    &&&&x&=&0 \\
    &&&&x&=&4
    \end{array}\)
12. \(\begin{array}{rrrrrrr}
    \\ \\ \\ \\ \\ \\
    v^2&-&8v&-&3&=&-3 \\
    &&&+&3&&+3 \\
    \midrule
    &&v^2&-&8v&=&0 \\
    &&v(v&-&8)&=&0 \\ \\
    &&&&v&=&0 \\
    &&&&v&=&8
    \end{array}\)
13. \(\begin{array}{rrrrrrrr}
    \\ \\ \\ \\ \\ \\
    &x^2&-&5x&-&1&=&-5 \\
    &&&&+&5&&+5 \\
    \midrule
    &x^2&-&5x&+&4&=&0 \\
    (x&-&4)&(x&-&1)&=&0 \\ \\
    &&&&&x&=&4 \\
    &&&&&x&=&1
    \end{array}\)
14. \(\begin{array}{rrrrrrrr}
    \\ \\ \\ \\ \\ \\
    &a^2&-&6a&+&6&=&-2 \\
    &&&&+&2&=&+2 \\
    \midrule
    &a^2&-&6a&+&8&=&0 \\
    (a&-&4)&(a&-&2)&=&0 \\ \\
    &&&&&a&=&4 \\
    &&&&&a&=&2
    \end{array}\)
15. \(\begin{array}{rrrrrrrr}
    \\ \\ \\ \\ \\ \\ \\ \\
    &7x^2&+&17x&-&20&=&-8 \\
    &&&&+&8&&+8 \\
    \midrule
    &7x^2&+&17x&-&12&=&0 \\
    (7x&-&4)&(x&+&3)&=&0 \\ \\
    &&&&&x&=&\dfrac{4}{7} \\ \\
    &&&&&x&=&-3
    \end{array}\)
16. \(\begin{array}{rrrrrrrr}
    \\ \\ \\ \\ \\ \\ \\ \\
    &4n^2&-&13n&+&8&=&5 \\
    &&&&-&5&&-5 \\
    \midrule
    &4n^2&-&13n&+&3&=&0 \\
    (4n&-&1)&(n&-&3)&=&0 \\ \\
    &&&&&n&=&\dfrac{1}{4} \\ \\
    &&&&&n&=&3
    \end{array}\)
17. \(\begin{array}{rrrrrrrr}
    \\ \\ \\ \\ \\ \\
    &x^2&-&6x&&&=&16 \\
    &&&&-&16&&-16 \\
    \midrule
    &x^2&-&6x&-&16&=&0 \\
    (x&-&8)&(x&+&2)&=&0 \\ \\
    &&&&&x&=&8 \\
    &&&&&x&=&-2 \\
    \end{array}\)
18. \(\begin{array}{rrrrr}
    \\ \\ \\ \\
    7n^2&-&28n&=&0 \\
    7n(n&-&4)&=&0 \\ \\
    &&n&=&0 \\
    &&n&=&4
    \end{array}\)
19. \(\begin{array}{rcrrrrrrrr}
    \\ \\ \\ \\ \\ \\ \\
    &4k^2&+&22k&+&23&=&6k&+&7 \\
    &&-&6k&-&7&&-6k&-&7 \\
    \midrule
    &4k^2&+&16k&+&16&=&0&& \\
    &4(k^2&+&4k&+&4)&=&0&& \\
    4(k&+&2)&(k&+&2)&=&0&& \\ \\
    &&&&&k&=&-2&&
    \end{array}\)
20. \(\begin{array}{rrrrrrrrrr}
    \\ \\ \\ \\ \\ \\
    &a^2&+&7a&-&9&=&-3&+&6a \\
    &&-&6a&+&3&&+3&-&6a \\
    \midrule
    &a^2&+&a&-&6&=&0&& \\
    (a&+&3)&(a&-&2)&=&0&& \\ \\
    &&&&&a&=&-3&& \\
    &&&&&a&=&2&&
    \end{array}\)
21. \(\begin{array}{rrrrrrrrrrrr}
    \\ \\ \\ \\ \\ \\
    &9x^2&-&46&+&7x&=&7x&+&8x^2&+&3 \\
    &-8x^2&-&3&-&7x&&-7x&-&8x^2&-&3 \\
    \midrule
    &&&x^2&-&49&=&0&&&& \\
    (x&-&7)&(x&+&7)&=&0&&&& \\ \\
    &&&&&x&=&7&&&& \\
    &&&&&x&=&-7&&&&
    \end{array}\)
22. \(\begin{array}{rrrrrrrr}
    \\ \\ \\ \\ \\ \\
    &x^2&+&10x&+&30&=&6 \\
    &&&&-&6&=&-6 \\
    \midrule
    &x^2&+&10x&+&24&=&0 \\
    (x&+&6)&(x&+&4)&=&0 \\ \\
    &&&&&x&=&-6 \\
    &&&&&x&=&-4
    \end{array}\)
23. \(\begin{array}{rrrrrrrrrr}
    \\ \\ \\ \\ \\ \\ \\ \\
    &40p^2&+&183p&-&168&=&p&+&5p^2 \\
    &-5p^2&-&p&&&&-p&-&5p^2 \\
    \midrule
    &35p^2&+&182p&-&168&=&0&& \\
    &7(5p^2&+&26p&-&24)&=&0&& \\
    7(p&+&6)&(5p&-&4)&=&0&& \\ \\
    &&&&&p&=&-6&& \\ \\
    &&&&&p&=&\dfrac{4}{5}&&
    \end{array}\)
24. \(\begin{array}{rcrrrrrr}
    \\ \\ \\ \\ \\ \\ \\ \\ \\
    &24x^2&+&11x&-&80&=&3x \\
    &&-&3x&&&&-3x \\
    \midrule
    &24x^2&+&8x&-&80&=&0 \\
    &8(3x^2&+&x&-&10)&=&0 \\
    8(3x&-&5)&(x&+&2)&=&0 \\ \\
    &&&&&x&=&\dfrac{5}{3} \\ \\
    &&&&&x&=&-2
    \end{array}\)