Answer Key 10.8

$x=5, x=2$
$x-5=0, x-2=0$

$0=(x-5)(x-2)$
$0=x^2-7x+10$
$x=3, x=6$
$x-3=0, x-6=0$

$(x-3)(x-6)=0$
$0=x^2-9x+18$
$x=20, x=2$
$x-20=0, x-2=0$

$0=(x-20)(x-2)$
$0=x^2-22x+40$
$x=13, x=1$
$x-13=0, x-1=0$

$0=(x-13)(x-1)$
$0=x^2-14x+13$
$x=4, x=4$
$x-4=0, x-4=0$

$0=(x-4)(x-4)$
$0=x^2-8x+16$
$x=0, x=9$
$x-9=0, x$

$x(x-9)=0$
$0=x^2-9x$
$x=\dfrac{3}{4}, x=\dfrac{1}{4} \\$
$x-\dfrac{3}{4}=0, x-\dfrac{1}{4}=0$

$0=\left(x-\dfrac{3}{4}\right)\left(x-\dfrac{1}{4}\right) \\$
$0=x^2-x+\dfrac{3}{16}$
$x=\dfrac{5}{8}, x=\dfrac{5}{7} \\$
$x-\dfrac{5}{8}=0, x-\dfrac{5}{7}=0$

$0=\left(x-\dfrac{5}{8}\right)\left(x-\dfrac{5}{7}\right) \\$
$0=x^2-\dfrac{75}{56}x+\dfrac{25}{56}$
$x=\dfrac{1}{2}, x=\dfrac{1}{3} \\$
$x-\dfrac{1}{2}=0, x-\dfrac{1}{3}=0$

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

$0=\left(x-\dfrac{1}{2}\right)\left(x-\dfrac{2}{3}\right) \\$
$0=x^2-\dfrac{7}{6}x+\dfrac{1}{3}$
$x=5, x=-5$
$x-5=0, x+5=0$

$0=(x-5)(x+5)$
$0=x^2-25$
$x=1, x=-1$
$x-1=0, x+1=0$

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

$0=(x-\dfrac{1}{5})(x+\dfrac{1}{5}) \\$
$0=x^2-\dfrac{1}{25}$
$x=\sqrt{7}, x=-\sqrt{7}$
$x-\sqrt{7}=0, x+\sqrt{7}=0$

$0=(x-\sqrt{7})(x+\sqrt{7})$
$0=x^2-7$
$x=\sqrt{11}, x=-\sqrt{11}$
$x-\sqrt{11}=0, x+\sqrt{11}=0$

$0=(x-\sqrt{11})(x+\sqrt{11})$
$0=x^2-11$
$x=2\sqrt{3}, x=-2\sqrt{3}$
$x-2\sqrt{3}=0, x+2\sqrt{3}=0$

$0=(x-2\sqrt{3})(x+2\sqrt{3})$
$0=x^2-12$
$x=3, x=5, x=8$
$(x-3)=0, (x-5)=0, (x-8)=0$

$\begin{array}{rrcrcrrrr} (x&-&3)(x&-&5)(x&-&8)&=&0 \\ (x^2&-&8x&+&15)(x&-&8)&=&0 \\ x^3&-&8x^2&+&15x&&&& \\ &-&8x^2&+&64x&-&120&=&0 \\ \midrule x^3&-&16x^2&+&79x&-&120&=&0 \end{array}$
$x=-4, x=0, x=4$
$x+4=0, x, x-4=0$

$x(x+4)(x-4)=0$
$x(x^2-16)=0$
$x^3-16x=0$
$x=-9, x+6=0, x=-2$
$x+9=0, x+6=0, x+2=0$

$\begin{array}{rrcrcrrrr} (x&+&9)(x&+&6)(x&+&2)&=&0 \\ (x^2&+&15x&+&54)(x&+&2)&=&0 \\ x^3&+&15x^2&+&54x&&&& \\ &+&2x^2&+&30x&+&108&=&0 \\ \midrule x^3&+&17x^2&+&84x&+&108&=&0 \end{array}$
$x=-1, x=1, x=5$
$x+1=0, x-1=0, x-5=0$

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

$(x+2)(x-2)(x-5)(x+5)=0$
$(x^2-4)(x^2-25)=0$
$x^4-29x^2+100=0$
$x=2\sqrt{3}, x=-2\sqrt{3}, x=\sqrt{5}, x=-\sqrt{5}$
$x-2\sqrt{3}=0, x+2\sqrt{3}=0, x-\sqrt{5}=0, x+\sqrt{5}=0$

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

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