Answer Key 10.8

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1. \(x=5, x=2\)
   \(x-5=0, x-2=0\)

   \(0=(x-5)(x-2)\)
   \(0=x^2-7x+10\)

2. \(x=3, x=6\)
   \(x-3=0, x-6=0\)

   \((x-3)(x-6)=0\)
   \(0=x^2-9x+18\)

3. \(x=20, x=2\)
   \(x-20=0, x-2=0\)

   \(0=(x-20)(x-2)\)
   \(0=x^2-22x+40\)

4. \(x=13, x=1\)
   \(x-13=0, x-1=0\)

   \(0=(x-13)(x-1)\)
   \(0=x^2-14x+13\)

5. \(x=4, x=4\)
   \(x-4=0, x-4=0\)

   \(0=(x-4)(x-4)\)
   \(0=x^2-8x+16\)

6. \(x=0, x=9\)
   \(x-9=0, x\)

   \(x(x-9)=0\)
   \(0=x^2-9x\)

7. \(x=\dfrac{3}{4}, x=\dfrac{1}{4} \\ \)
   \(x-\dfrac{3}{4}=0, x-\dfrac{1}{4}=0 \)

   \(0=\left(x-\dfrac{3}{4}\right)\left(x-\dfrac{1}{4}\right) \\ \)
   \(0=x^2-x+\dfrac{3}{16}\)

8. \(x=\dfrac{5}{8}, x=\dfrac{5}{7} \\ \)
   \(x-\dfrac{5}{8}=0, x-\dfrac{5}{7}=0\)

   \(0=\left(x-\dfrac{5}{8}\right)\left(x-\dfrac{5}{7}\right) \\ \)
   \(0=x^2-\dfrac{75}{56}x+\dfrac{25}{56}\)

9. \(x=\dfrac{1}{2}, x=\dfrac{1}{3} \\ \)
   \(x-\dfrac{1}{2}=0, x-\dfrac{1}{3}=0\)

   \(0=\left(x-\dfrac{1}{2}\right)\left(x-\dfrac{1}{3}\right) \\ \)
   \(0=x^2-\dfrac{5}{6}x+\dfrac{1}{6}\)

10. \(x=\dfrac{1}{2}, x=\dfrac{2}{3} \\ \)
    \(x-\dfrac{1}{2}=0, x-\dfrac{2}{3}=0\)

    \(0=\left(x-\dfrac{1}{2}\right)\left(x-\dfrac{2}{3}\right) \\ \)
    \(0=x^2-\dfrac{7}{6}x+\dfrac{1}{3}\)

11. \(x=5, x=-5\)
    \(x-5=0, x+5=0\)

    \(0=(x-5)(x+5)\)
    \(0=x^2-25\)

12. \(x=1, x=-1\)
    \(x-1=0, x+1=0\)

    \(0=(x-1)(x+1)\)
    \(0=x^2-1\)

13. \(x=\dfrac{1}{5}, x=-\dfrac{1}{5} \\ \)
    \(x-\dfrac{1}{5}=0, x+\dfrac{1}{5}=0\)

    \(0=(x-\dfrac{1}{5})(x+\dfrac{1}{5}) \\ \)
    \(0=x^2-\dfrac{1}{25}\)

14. \(x=\sqrt{7}, x=-\sqrt{7}\)
    \(x-\sqrt{7}=0, x+\sqrt{7}=0\)

    \(0=(x-\sqrt{7})(x+\sqrt{7})\)
    \(0=x^2-7\)

15. \(x=\sqrt{11}, x=-\sqrt{11}\)
    \(x-\sqrt{11}=0, x+\sqrt{11}=0\)

    \(0=(x-\sqrt{11})(x+\sqrt{11})\)
    \(0=x^2-11\)

16. \(x=2\sqrt{3}, x=-2\sqrt{3}\)
    \(x-2\sqrt{3}=0, x+2\sqrt{3}=0\)

    \(0=(x-2\sqrt{3})(x+2\sqrt{3})\)
    \(0=x^2-12\)

17. \(x=3, x=5, x=8\)
    \((x-3)=0, (x-5)=0, (x-8)=0\)

    \(\begin{array}{rrcrcrrrr}
    (x&-&3)(x&-&5)(x&-&8)&=&0 \\
    (x^2&-&8x&+&15)(x&-&8)&=&0 \\
    x^3&-&8x^2&+&15x&&&& \\
    &-&8x^2&+&64x&-&120&=&0 \\
    \midrule
    x^3&-&16x^2&+&79x&-&120&=&0
    \end{array}\)

18. \(x=-4, x=0, x=4\)
    \(x+4=0, x, x-4=0\)

    \(x(x+4)(x-4)=0\)
    \(x(x^2-16)=0\)
    \(x^3-16x=0\)

19. \(x=-9, x+6=0, x=-2\)
    \(x+9=0, x+6=0, x+2=0\)

    \(\begin{array}{rrcrcrrrr}
    (x&+&9)(x&+&6)(x&+&2)&=&0 \\
    (x^2&+&15x&+&54)(x&+&2)&=&0 \\
    x^3&+&15x^2&+&54x&&&& \\
    &+&2x^2&+&30x&+&108&=&0 \\
    \midrule
    x^3&+&17x^2&+&84x&+&108&=&0
    \end{array}\)

20. \(x=-1, x=1, x=5\)
    \(x+1=0, x-1=0, x-5=0\)

    \((x+1)(x-1)(x-5)=0\)
    \((x^2-1)(x-5)=0\)
    \(x^3-5x^2-x+5=0\)

21. \(x=-2, x=2, x=5, x=-5\)
    \(x+2=0, x-2=0, x-5=0, x+5=0\)

    \((x+2)(x-2)(x-5)(x+5)=0\)
    \((x^2-4)(x^2-25)=0\)
    \(x^4-29x^2+100=0\)

22. \(x=2\sqrt{3}, x=-2\sqrt{3}, x=\sqrt{5}, x=-\sqrt{5}\)
    \(x-2\sqrt{3}=0, x+2\sqrt{3}=0, x-\sqrt{5}=0, x+\sqrt{5}=0\)

    \((x-2\sqrt{3})(x+2\sqrt{3})(x-\sqrt{5})(x+\sqrt{5})=0\)
    \((x^2-12)(x^2-5)=0\)
    \(x^4-17x^2+60=0\)