Answer Key 10.3
[latexpage]
- \(\begin{array}{rrl}
\\ \\ \\
\dfrac{30}{2}&=&15 \\ \\
15^2&=&225 \\
\therefore x^2&-&30x+225\text{ or }(x-15)^2
\end{array}\) - \(\begin{array}{rrl}
\\ \\ \\
\dfrac{24}{2}&=&12 \\ \\
12^2&=&144 \\
\therefore a^2&-&24a+144\text{ or }(a-12)^2
\end{array}\) - \(\begin{array}{rrl}
\\ \\ \\
\dfrac{36}{2}&=&18 \\ \\
18^2&=&324 \\
\therefore m^2&-&36m+324\text{ or }(m-18)^2
\end{array}\) - \(\begin{array}{rrl}
\\ \\ \\
\dfrac{34}{2}&=&17 \\ \\
17^2&=&289 \\
\therefore x^2&-&34x+289\text{ or }(x-17)^2
\end{array}\) - \(\begin{array}{rrl}
\\ \\ \\ \\
\dfrac{15}{2}&=&7.5 \\ \\
7.5^2&=&56.25 \\
\therefore x^2&-&15x+56.25\text{ or }\left(x-\dfrac{15}{2}\right)^2
\end{array}\) - \(\begin{array}{rrl}
\\ \\ \\ \\ \\
\dfrac{19}{2}&=&\dfrac{19}{2} \\ \\
\left(\dfrac{19}{2}\right)^2&=&\dfrac{361}{4} \\
\therefore r^2&-&19r+\dfrac{361}{4}\text{ or } \left(r-\dfrac{19}{2}\right)^2
\end{array}\) - \(\begin{array}{rrl}
\\ \\ \\ \\ \\
\dfrac{1}{2}&& \\ \\
\left(\dfrac{1}{2}\right)^2&=&\dfrac{1}{4} \\
\therefore y^2&-&y+\dfrac{1}{4}\text{ or } \left(y-\dfrac{1}{2}\right)^2
\end{array}\) - \(\begin{array}{rrl}
\\ \\ \\ \\ \\
\dfrac{17}{2}&& \\ \\
\left(\dfrac{17}{2}\right)^2&=&\dfrac{289}{4} \\
\therefore p^2&-&17p+\dfrac{289}{4}\text{ or }\left(p-\dfrac{17}{2}\right)^2
\end{array}\) - \(\begin{array}{rrrrrlrrr}
\\ \\ \\ \\ \\ \\
x^2&-&16x&+&55&=&0&& \\
&&&-&55&&-55&& \\
\midrule
&&x^2&-&16x&=&-55&& \\ \\
x^2&-&16x&+&64&=&64&-&55 \\
&&(x&-&8)^2&=&9&&
\end{array}\)\(\sqrt{(x-8)^2}=\sqrt{9}\)
\(\begin{array}{rrrrrrr}
x&-&8&=&\pm &3& \\
&+&8&&+ &8& \\
\midrule
&&x&=&8&\pm &3 \\
&&x&=&5,&11 &
\end{array}\) - \(\begin{array}{rrrrrrrrr}
\\ \\ \\ \\ \\
n^2&-&4n&-&12&=&0&& \\
&&&+&12&&+12&& \\
\midrule
&&n^2&-&4n&=&12&& \\ \\
n^2&-&4n&+&4&=&12&+&4 \\
&&(n&-&2)^2&=&16&&
\end{array}\)\(\sqrt{(n-2)^2}=\pm \sqrt{16}\)
\(\begin{array}{rrrrrrr}
n&-&2&=&\pm &4& \\
&+&2&&+&2& \\
\midrule
&&n&=&2&\pm &4 \\
&&n&=&6,&-2&
\end{array}\) - \(\begin{array}{rrrrrrrrr}
\\ \\ \\ \\ \\
v^2&-&4v&-&21&=&0&& \\
&&&+&21&&+21&& \\
\midrule
&&v^2&-&4v&=&21&& \\ \\
v^2&-&4v&+&4&=&21&+&4 \\
&&(v&-&2)^2&=&25&&
\end{array}\)\(\sqrt{(v-2)^2}=\sqrt{25}\)
\(\begin{array}{rrrrrrr}
v&-&2&=&\pm &5& \\
&+&2&&+&2& \\
\midrule
&&v&=&2&\pm &5 \\
&&v&=&7,&-3&
\end{array}\) - \(\begin{array}{rrrrrrrrr}
\\ \\ \\ \\ \\
b^2&+&8b&+&7&=&0&& \\
&&&-&7&&-7&& \\
\midrule
&&b^2&+&8b&=&-7&& \\ \\
b^2&+&8b&+&16&=&-7&+&16 \\
&&(b&+&4)^2&=&9&&
\end{array}\)\(\sqrt{(b+4)^2}=\sqrt{9}\)
\(\begin{array}{rrrrrrr}
b&+&4&=&\pm&3& \\
&-&4&&-&4& \\
\midrule
&&b&=&-4&\pm &3 \\
&&b&=&-7,&-1&
\end{array}\) - \(\begin{array}{rrrrrrrrr}
\\
x^2&-&8x&+&16&=&-6&+&16 \\
&&(x&-&4)^2&=&10&&
\end{array}\)\(\sqrt{(x-4)^2}=\sqrt{10}\)
\(\begin{array}{rrrrrrr}
x&-&4&=&\pm&\sqrt{10}& \\
&+&4&&+&4& \\
\midrule
&&x&=&4&\pm&\sqrt{10}
\end{array}\) - \(\begin{array}{rrrrrrrrr}
\\ \\ \\
x^2&&&-&13&=&4x&& \\
&-&4x&+&13&&-4x&+&13 \\
\midrule
x^2&-&4x&+&4&=&13&+&4 \\
&&(x&-&2)^2&=&17&&
\end{array}\)\(\sqrt{(x-2)^2}=\sqrt{17}\)
\(\begin{array}{rrrrrrr}
x&-&2&=&\pm&\sqrt{17}& \\
&+&2&&+&2& \\
\midrule
&&x&=&2&\pm&\sqrt{17}
\end{array}\) - \(\begin{array}{rrrrrrrrr}
\\ \\ \\ \\ \\ \\ \\ \\
&&\dfrac{3}{3}(k^2&+&8k)&=&\dfrac{-1}{3}&& \\ \\
&&k^2&+&8k&=&-\dfrac{1}{3}&& \\ \\
k^2&+&8k&+&16&=&-\dfrac{1}{3}&+&16 \\ \\
&&(k&+&4)^2&=&15\dfrac{2}{3}&&
\end{array}\)\(\sqrt{(k+4)^2}=\sqrt{15\dfrac{2}{3}}\)
\(\begin{array}{rrrrrrr}
k&+&4&=&\pm &\sqrt{\dfrac{47}{3}}& \\
&-&4&&-&4& \\
\midrule
&&k&=&-4&\pm &\sqrt{\dfrac{47}{3}}
\end{array}\) - \(\begin{array}{rrrrrrrrr}
\\ \\ \\ \\ \\
&&\dfrac{4}{4}(a^2&+&9a)&=&\dfrac{-2}{4}&& \\ \\
a^2&+&9a&+&20.25&=&-\dfrac{1}{2}&+&20.25 \\ \\
&&(a&+&4.5)^2&=&19.75&&
\end{array}\)\(\sqrt{(a+4.5)^2}=\pm \sqrt{19.75}\)
\(\begin{array}{rrrrrcl}
a&+&4.5&=&\pm&\sqrt{19.75}& \\
&-&4.5&&-&4.5& \\
\midrule
&&a&=&-4.5&\pm&\sqrt{19.75}
\end{array}\)
