Answer Key 10.3

  1. \begin{array}{rrl} \\ \\ \\ \dfrac{30}{2}&=&15 \\ \\ 15^2&=&225 \\ \therefore x^2&-&30x+225\text{ or }(x-15)^2 \end{array}
  2. \begin{array}{rrl} \\ \\ \\ \dfrac{24}{2}&=&12 \\ \\ 12^2&=&144 \\ \therefore a^2&-&24a+144\text{ or }(a-12)^2 \end{array}
  3. \begin{array}{rrl} \\ \\ \\ \dfrac{36}{2}&=&18 \\ \\ 18^2&=&324 \\ \therefore m^2&-&36m+324\text{ or }(m-18)^2 \end{array}
  4. \begin{array}{rrl} \\ \\ \\ \dfrac{34}{2}&=&17 \\ \\ 17^2&=&289 \\ \therefore x^2&-&34x+289\text{ or }(x-17)^2 \end{array}
  5. \begin{array}{rrl} \\ \\ \\ \\ \dfrac{15}{2}&=&7.5 \\ \\ 7.5^2&=&56.25 \\ \therefore x^2&-&15x+56.25\text{ or }\left(x-\dfrac{15}{2}\right)^2 \end{array}
  6. \begin{array}{rrl} \\ \\ \\ \\ \\ \dfrac{19}{2}&=&\dfrac{19}{2} \\ \\ \left(\dfrac{19}{2}\right)^2&=&\dfrac{361}{4} \\ \therefore r^2&-&19r+\dfrac{361}{4}\text{ or } \left(r-\dfrac{19}{2}\right)^2 \end{array}
  7. \begin{array}{rrl} \\ \\ \\ \\ \\ \dfrac{1}{2}&& \\ \\ \left(\dfrac{1}{2}\right)^2&=&\dfrac{1}{4} \\ \therefore y^2&-&y+\dfrac{1}{4}\text{ or } \left(y-\dfrac{1}{2}\right)^2 \end{array}
  8. \begin{array}{rrl} \\ \\ \\ \\ \\ \dfrac{17}{2}&& \\ \\ \left(\dfrac{17}{2}\right)^2&=&\dfrac{289}{4} \\ \therefore p^2&-&17p+\dfrac{289}{4}\text{ or }\left(p-\dfrac{17}{2}\right)^2 \end{array}
  9. \begin{array}{rrrrrlrrr} \\ \\ \\ \\ \\ \\ x^2&-&16x&+&55&=&0&& \\ &&&-&55&&-55&& \\ \midrule &&x^2&-&16x&=&-55&& \\ \\ x^2&-&16x&+&64&=&64&-&55 \\ &&(x&-&8)^2&=&9&& \end{array}

    \sqrt{(x-8)^2}=\sqrt{9}

    \begin{array}{rrrrrrr} x&-&8&=&\pm &3& \\ &+&8&&+ &8& \\ \midrule &&x&=&8&\pm &3 \\ &&x&=&5,&11 & \end{array}

  10. \begin{array}{rrrrrrrrr} \\ \\ \\ \\ \\ n^2&-&4n&-&12&=&0&& \\ &&&+&12&&+12&& \\ \midrule &&n^2&-&4n&=&12&& \\ \\ n^2&-&4n&+&4&=&12&+&4 \\ &&(n&-&2)^2&=&16&& \end{array}

    \sqrt{(n-2)^2}=\pm \sqrt{16}

    \begin{array}{rrrrrrr} n&-&2&=&\pm &4& \\ &+&2&&+&2& \\ \midrule &&n&=&2&\pm &4 \\ &&n&=&6,&-2& \end{array}

  11. \begin{array}{rrrrrrrrr} \\ \\ \\ \\ \\ v^2&-&4v&-&21&=&0&& \\ &&&+&21&&+21&& \\ \midrule &&v^2&-&4v&=&21&& \\ \\ v^2&-&4v&+&4&=&21&+&4 \\ &&(v&-&2)^2&=&25&& \end{array}

    \sqrt{(v-2)^2}=\sqrt{25}

    \begin{array}{rrrrrrr} v&-&2&=&\pm &5& \\ &+&2&&+&2& \\ \midrule &&v&=&2&\pm &5 \\ &&v&=&7,&-3& \end{array}

  12. \begin{array}{rrrrrrrrr} \\ \\ \\ \\ \\ b^2&+&8b&+&7&=&0&& \\ &&&-&7&&-7&& \\ \midrule &&b^2&+&8b&=&-7&& \\ \\ b^2&+&8b&+&16&=&-7&+&16 \\ &&(b&+&4)^2&=&9&& \end{array}

    \sqrt{(b+4)^2}=\sqrt{9}

    \begin{array}{rrrrrrr} b&+&4&=&\pm&3& \\ &-&4&&-&4& \\ \midrule &&b&=&-4&\pm &3 \\ &&b&=&-7,&-1& \end{array}

  13. \begin{array}{rrrrrrrrr} \\ x^2&-&8x&+&16&=&-6&+&16 \\ &&(x&-&4)^2&=&10&& \end{array}

    \sqrt{(x-4)^2}=\sqrt{10}

    \begin{array}{rrrrrrr} x&-&4&=&\pm&\sqrt{10}& \\ &+&4&&+&4& \\ \midrule &&x&=&4&\pm&\sqrt{10} \end{array}

  14. \begin{array}{rrrrrrrrr} \\ \\ \\ x^2&&&-&13&=&4x&& \\ &-&4x&+&13&&-4x&+&13 \\ \midrule x^2&-&4x&+&4&=&13&+&4 \\ &&(x&-&2)^2&=&17&& \end{array}

    \sqrt{(x-2)^2}=\sqrt{17}

    \begin{array}{rrrrrrr} x&-&2&=&\pm&\sqrt{17}& \\ &+&2&&+&2& \\ \midrule &&x&=&2&\pm&\sqrt{17} \end{array}

  15. \begin{array}{rrrrrrrrr} \\ \\ \\ \\ \\ \\ \\ \\ &&\dfrac{3}{3}(k^2&+&8k)&=&\dfrac{-1}{3}&& \\ \\ &&k^2&+&8k&=&-\dfrac{1}{3}&& \\ \\ k^2&+&8k&+&16&=&-\dfrac{1}{3}&+&16 \\ \\ &&(k&+&4)^2&=&15\dfrac{2}{3}&& \end{array}

    \sqrt{(k+4)^2}=\sqrt{15\dfrac{2}{3}}

    \begin{array}{rrrrrrr} k&+&4&=&\pm &\sqrt{\dfrac{47}{3}}& \\ &-&4&&-&4& \\ \midrule &&k&=&-4&\pm &\sqrt{\dfrac{47}{3}} \end{array}

  16. \begin{array}{rrrrrrrrr} \\ \\ \\ \\ \\ &&\dfrac{4}{4}(a^2&+&9a)&=&\dfrac{-2}{4}&& \\ \\ a^2&+&9a&+&20.25&=&-\dfrac{1}{2}&+&20.25 \\ \\ &&(a&+&4.5)^2&=&19.75&& \end{array}

    \sqrt{(a+4.5)^2}=\pm \sqrt{19.75}

    \begin{array}{rrrrrcl} a&+&4.5&=&\pm&\sqrt{19.75}& \\ &-&4.5&&-&4.5& \\ \midrule &&a&=&-4.5&\pm&\sqrt{19.75} \end{array}

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