Answer Key 10.8
- \(x=5, x=2\)
\(x-5=0, x-2=0\)\(0=(x-5)(x-2)\)
\(0=x^2-7x+10\) - \(x=3, x=6\)
\(x-3=0, x-6=0\)\((x-3)(x-6)=0\)
\(0=x^2-9x+18\) - \(x=20, x=2\)
\(x-20=0, x-2=0\)\(0=(x-20)(x-2)\)
\(0=x^2-22x+40\) - \(x=13, x=1\)
\(x-13=0, x-1=0\)\(0=(x-13)(x-1)\)
\(0=x^2-14x+13\) - \(x=4, x=4\)
\(x-4=0, x-4=0\)\(0=(x-4)(x-4)\)
\(0=x^2-8x+16\) - \(x=0, x=9\)
\(x-9=0, x\)\(x(x-9)=0\)
\(0=x^2-9x\) - \(x=\dfrac{3}{4}, x=\dfrac{1}{4} \\ \)
\(x-\dfrac{3}{4}=0, x-\dfrac{1}{4}=0 \)\(0=\left(x-\dfrac{3}{4}\right)\left(x-\dfrac{1}{4}\right) \\ \)
\(0=x^2-x+\dfrac{3}{16}\) - \(x=\dfrac{5}{8}, x=\dfrac{5}{7} \\ \)
\(x-\dfrac{5}{8}=0, x-\dfrac{5}{7}=0\)\(0=\left(x-\dfrac{5}{8}\right)\left(x-\dfrac{5}{7}\right) \\ \)
\(0=x^2-\dfrac{75}{56}x+\dfrac{25}{56}\) - \(x=\dfrac{1}{2}, x=\dfrac{1}{3} \\ \)
\(x-\dfrac{1}{2}=0, x-\dfrac{1}{3}=0\)\(0=\left(x-\dfrac{1}{2}\right)\left(x-\dfrac{1}{3}\right) \\ \)
\(0=x^2-\dfrac{5}{6}x+\dfrac{1}{6}\) - \(x=\dfrac{1}{2}, x=\dfrac{2}{3} \\ \)
\(x-\dfrac{1}{2}=0, x-\dfrac{2}{3}=0\)\(0=\left(x-\dfrac{1}{2}\right)\left(x-\dfrac{2}{3}\right) \\ \)
\(0=x^2-\dfrac{7}{6}x+\dfrac{1}{3}\) - \(x=5, x=-5\)
\(x-5=0, x+5=0\)\(0=(x-5)(x+5)\)
\(0=x^2-25\) - \(x=1, x=-1\)
\(x-1=0, x+1=0\)\(0=(x-1)(x+1)\)
\(0=x^2-1\) - \(x=\dfrac{1}{5}, x=-\dfrac{1}{5} \\ \)
\(x-\dfrac{1}{5}=0, x+\dfrac{1}{5}=0\)\(0=(x-\dfrac{1}{5})(x+\dfrac{1}{5}) \\ \)
\(0=x^2-\dfrac{1}{25}\) - \(x=\sqrt{7}, x=-\sqrt{7}\)
\(x-\sqrt{7}=0, x+\sqrt{7}=0\)\(0=(x-\sqrt{7})(x+\sqrt{7})\)
\(0=x^2-7\) - \(x=\sqrt{11}, x=-\sqrt{11}\)
\(x-\sqrt{11}=0, x+\sqrt{11}=0\)\(0=(x-\sqrt{11})(x+\sqrt{11})\)
\(0=x^2-11\) - \(x=2\sqrt{3}, x=-2\sqrt{3}\)
\(x-2\sqrt{3}=0, x+2\sqrt{3}=0\)\(0=(x-2\sqrt{3})(x+2\sqrt{3})\)
\(0=x^2-12\) - \(x=3, x=5, x=8\)
\((x-3)=0, (x-5)=0, (x-8)=0\)\(\begin{array}{rrcrcrrrr}
(x&-&3)(x&-&5)(x&-&8)&=&0 \\
(x^2&-&8x&+&15)(x&-&8)&=&0 \\
x^3&-&8x^2&+&15x&&&& \\
&-&8x^2&+&64x&-&120&=&0 \\
\hline
x^3&-&16x^2&+&79x&-&120&=&0
\end{array}\) - \(x=-4, x=0, x=4\)
\(x+4=0, x, x-4=0\)\(x(x+4)(x-4)=0\)
\(x(x^2-16)=0\)
\(x^3-16x=0\) - \(x=-9, x+6=0, x=-2\)
\(x+9=0, x+6=0, x+2=0\)\(\begin{array}{rrcrcrrrr}
(x&+&9)(x&+&6)(x&+&2)&=&0 \\
(x^2&+&15x&+&54)(x&+&2)&=&0 \\
x^3&+&15x^2&+&54x&&&& \\
&+&2x^2&+&30x&+&108&=&0 \\
\hline
x^3&+&17x^2&+&84x&+&108&=&0
\end{array}\) - \(x=-1, x=1, x=5\)
\(x+1=0, x-1=0, x-5=0\)\((x+1)(x-1)(x-5)=0\)
\((x^2-1)(x-5)=0\)
\(x^3-5x^2-x+5=0\) - \(x=-2, x=2, x=5, x=-5\)
\(x+2=0, x-2=0, x-5=0, x+5=0\)\((x+2)(x-2)(x-5)(x+5)=0\)
\((x^2-4)(x^2-25)=0\)
\(x^4-29x^2+100=0\) - \(x=2\sqrt{3}, x=-2\sqrt{3}, x=\sqrt{5}, x=-\sqrt{5}\)
\(x-2\sqrt{3}=0, x+2\sqrt{3}=0, x-\sqrt{5}=0, x+\sqrt{5}=0\)\((x-2\sqrt{3})(x+2\sqrt{3})(x-\sqrt{5})(x+\sqrt{5})=0\)
\((x^2-12)(x^2-5)=0\)
\(x^4-17x^2+60=0\)
