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Solve each of the following quadratic equations using the method of extraction of roots.
$\begin{array}{cccccc}{x}^{2}-49\hfill & =\hfill & 0.\hfill & \text{Rewrite}\text{.}\hfill & \hfill & \hfill \\ \hfill {x}^{2}& =\hfill & 49\hfill & \hfill & \hfill & \hfill \\ \hfill x& =\hfill & \pm \sqrt{49}\hfill & \hfill & \hfill & \hfill \\ \hfill x& =\hfill & \pm 7\hfill & \hfill & \hfill & \hfill \\ Check:\hfill & \hfill & {(7)}^{2}=49\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill & {(-7)}^{2}=49\hfill & \text{Is this correct}\hfill \\ \hfill & \hfill & 49=49\hfill & \text{Yes,\hspace{0.17em}this\hspace{0.17em}is\hspace{0.17em}correct}\text{.}\hfill & 49=49\hfill & \text{Yes,\hspace{0.17em}this\hspace{0.17em}is\hspace{0.17em}correct}\text{.}\hfill \end{array}$
$\begin{array}{rrr}25{a}^{2}& =& 36\hfill \\ {a}^{2}& =& \frac{36}{25}\hfill \\ a& =& \pm \sqrt{\frac{36}{25}}\hfill \\ a& =& \pm \frac{6}{5}\hfill \end{array}$
$\begin{array}{lllllllllll}Check:\hfill & \hfill & \hfill 25{(\frac{6}{5})}^{2}& =\hfill & 36\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill & \hfill & \hfill 25{(\frac{-6}{5})}^{2}& =\hfill & 36\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill \\ \hfill & \hfill & \hfill 25{(\frac{36}{25})}^{2}& =\hfill & 36\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill & \hfill & \hfill 25(\frac{36}{25})& =\hfill & 36\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill \\ \hfill & \hfill & \hfill 36& =\hfill & 36\hfill & \text{Yes,\hspace{0.17em}this\hspace{0.17em}is\hspace{0.17em}correct}\text{.}\hfill & \hfill & \hfill 36& =\hfill & 36\hfill & \text{Yes,\hspace{0.17em}this\hspace{0.17em}is\hspace{0.17em}correct}\text{.}\hfill \end{array}$
$\begin{array}{rrr}4{m}^{2}-32& =& 0\hfill \\ 4{m}^{2}& =\hfill & 32\hfill \\ {m}^{2}& =& \frac{32}{4}\hfill \\ {m}^{2}& =& 8\hfill \\ m& =& \pm \sqrt{8}\hfill \\ m& =& \pm 2\sqrt{2}\hfill \end{array}$
$\begin{array}{llllllllll}\hfill Check:& \hfill & \hfill 4{(2\sqrt{2})}^{2}& =\hfill & 32\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill & \hfill 4{(-2\sqrt{2})}^{2}& =\hfill & 32\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill \\ \hfill & \hfill & \hfill 4[{2}^{2}{(\sqrt{2})}^{2}]& =\hfill & 32\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill & \hfill 4[{(-2)}^{2}{(\sqrt{2})}^{2}]& =\hfill & 32\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill \\ \hfill & \hfill & \hfill 4[4\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}2]& =\hfill & 32\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill & \hfill 4[4\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}2]& =\hfill & 32\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill \\ \hfill & \hfill & \hfill 4\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}8& =\hfill & 32\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill & \hfill 4\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}8& =\hfill & 32\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill \\ \hfill & \hfill & \hfill 32& =\hfill & 32\hfill & \text{Yes,\hspace{0.17em}this\hspace{0.17em}is\hspace{0.17em}correct}\text{.}\hfill & \hfill 32& =\hfill & 32\hfill & \text{Yes,\hspace{0.17em}this\hspace{0.17em}is\hspace{0.17em}correct}\text{.}\hfill \end{array}$
Solve
$5{x}^{2}-15{y}^{2}{z}^{7}=0$ for
$x.$
$\begin{array}{lllll}5{x}^{2}& =& 15{y}^{2}{z}^{7}& & \text{Divide\hspace{0.17em}both\hspace{0.17em}sides\hspace{0.17em}by\hspace{0.17em}5}\text{.}\\ \hfill {x}^{2}& =& 3{y}^{2}{z}^{7}& & \\ \hfill x& =& \pm \sqrt{3{y}^{2}{z}^{7}}& & \\ \hfill x& =& \pm y{z}^{3}\sqrt{3z}& & \end{array}$
Calculator problem. Solve
$14{a}^{2}-235=0.$ Round to the nearest hundredth.
$\begin{array}{ccccc}14{a}^{2}-235\hfill & =& 0.\hfill & & \text{Rewrite}\text{.}\hfill \\ \hfill 14{a}^{2}& =& 235& & \text{Divide\hspace{0.17em}both\hspace{0.17em}sides\hspace{0.17em}by\hspace{0.17em}14}\text{.}\\ \hfill {a}^{2}& =& \frac{235}{14}& & \end{array}$
$\text{On\hspace{0.17em}the\hspace{0.17em}Calculator}$
$\begin{array}{ccc}\text{Type}& & 235\\ \text{Press}& & \begin{array}{|c|}\hline \xf7\\ \hline\end{array}\\ \text{Type}& & 14\\ \text{Press}& & \begin{array}{|c|}\hline =\\ \hline\end{array}\\ \text{Press}& & \surd \\ \text{Display\hspace{0.17em}reads:}& & 4.0970373\end{array}$
Rounding to the nearest hundredth produces 4.10. We must be sure to insert the
$\pm $ symbol.
$a\approx \pm 4.10$
$\begin{array}{lll}{k}^{2}& =& -64\\ k& =& \pm \sqrt{-64}\end{array}$
The radicand is
negative so no real number solutions exist.
Solve each of the following quadratic equations using the method of extraction of roots.
${x}^{2}-144=0$
$x=\pm 12$
$9{y}^{2}-121=0$
$y=\pm \frac{11}{3}$
$6{a}^{2}=108$
$a=\pm 3\sqrt{2}$
Solve $4{n}^{2}=24{m}^{2}{p}^{8}$ for $n.$
$n=\pm m{p}^{4}\sqrt{6}$
Solve $5{p}^{2}{q}^{2}=45{p}^{2}$ for $q.$
$q=\pm 3$
Solve $16{m}^{2}-2206=0.$ Round to the nearest hundredth.
$m=\pm 11.74$
${h}^{2}=-100$
Solve each of the following quadratic equations using the method of extraction of roots.
$$\begin{array}{lllllllll}\hfill {(x+2)}^{2}& =\hfill & 81\hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ \hfill x+2& =\hfill & \pm \sqrt{81}\hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ \hfill x+2& =\hfill & \pm 9\hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \text{Subtract\hspace{0.17em}}2\text{\hspace{0.17em}from\hspace{0.17em}both\hspace{0.17em}sides}\text{.}\hfill \\ \hfill x& =\hfill & -2\pm 9\hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ \hfill x& =\hfill & -2+9\hfill & \hfill & \text{and}\hfill & \hfill & \hfill x& =\hfill & -2-9\hfill \\ \hfill x& =\hfill & 7\hfill & \hfill & \hfill & \hfill & \hfill x& =\hfill & -11\hfill \end{array}$$
$$\begin{array}{lllll}{\left(a+3\right)}^{2}& =& 5& & \\ \hfill a+3& =& \pm \sqrt{5}& & \text{Subtract\hspace{0.17em}3\hspace{0.17em}from\hspace{0.17em}both\hspace{0.17em}sides}\text{.}\\ \hfill a& =& -3\pm \sqrt{5}& & \end{array}$$
Solve each of the following quadratic equations using the method of extraction of roots.
${\left(a+6\right)}^{2}=64$
$a=2,-14$
${\left(m-4\right)}^{2}=15$
$m=4\pm \sqrt{15}$
${\left(y-7\right)}^{2}=49$
$y=0,\text{\hspace{0.17em}}14$
${\left(k-1\right)}^{2}=12$
$k=1\pm 2\sqrt{3}$
${\left(x-11\right)}^{2}=0$
$x=11$
For the following problems, solve each of the quadratic equations using the method of extraction of roots.
${x}^{2}=36$
$x=\pm 6$
${x}^{2}=49$
${a}^{2}=9$
$a=\pm 3$
${a}^{2}=4$
${b}^{2}=1$
$b=\pm 1$
${a}^{2}=1$
${x}^{2}=25$
$x=\pm 5$
${x}^{2}=81$
${a}^{2}=5$
$a=\pm \sqrt{5}$
${a}^{2}=10$
${b}^{2}=12$
$b=\pm 2\sqrt{3}$
${b}^{2}=6$
${y}^{2}=3$
$y=\pm \sqrt{3}$
${y}^{2}=7$
${a}^{2}-8=0$
$a=\pm 2\sqrt{2}$
${a}^{2}-3=0$
${a}^{2}-5=0$
$a=\pm \sqrt{5}$
${y}^{2}-1=0$
${x}^{2}-10=0$
$x=\pm \sqrt{10}$
${x}^{2}-11=0$
$3{x}^{2}-27=0$
$x=\pm 3$
$5{b}^{2}-5=0$
$2{x}^{2}=50$
$x=\pm 5$
$4{a}^{2}=40$
$2{x}^{2}=24$
$x=\pm 2\sqrt{3}$
For the following problems, solve for the indicated variable.
${x}^{2}=4{a}^{2},$ for $x$
${x}^{2}=9{b}^{2},$ for $x$
$x=\pm 3b$
${a}^{2}=25{c}^{2},$ for $a$
${k}^{2}={m}^{2}{n}^{2},$ for $k$
$k=\pm mn$
${k}^{2}={p}^{2}{q}^{2}{r}^{2},$ for $k$
$2{y}^{2}=2{a}^{2}{n}^{2},$ for $y$
$y=\pm an$
$9{y}^{2}=27{x}^{2}{z}^{4},$ for $y$
${x}^{2}-{z}^{2}=0,$ for $x$
$x=\pm z$
${x}^{2}-{z}^{2}=0,$ for $z$
$5{a}^{2}-10{b}^{2}=0,$ for $a$
$a=b\sqrt{2},-b\sqrt{2}$
For the following problems, solve each of the quadratic equations using the method of extraction of roots.
${\left(x-1\right)}^{2}=4$
${\left(x-2\right)}^{2}=9$
$x=5,-1$
${\left(x-3\right)}^{2}=25$
${\left(a-5\right)}^{2}=36$
$x=11,-1$
${\left(a+3\right)}^{2}=49$
${\left(a+9\right)}^{2}=1$
$a=-8\text{\hspace{0.17em}},-10$
${\left(a-6\right)}^{2}=3$
${\left(x+4\right)}^{2}=5$
$a=-4\text{\hspace{0.17em}}\pm \sqrt{5}$
${\left(b+6\right)}^{2}=7$
${\left(x+1\right)}^{2}=a,$ for $x$
$x=-1\text{\hspace{0.17em}}\pm \sqrt{a}$
${\left(y+5\right)}^{2}=b,$ for $y$
${\left(y+2\right)}^{2}={a}^{2},$ for $y$
$y=-2\pm a$
${\left(x+10\right)}^{2}={c}^{2},$ for $x$
${\left(x-a\right)}^{2}={b}^{2},$ for $x$
$x=a\pm b$
${\left(x+c\right)}^{2}={a}^{2},$ for $x$
For the following problems, round each result to the nearest hundredth.
$8{a}^{2}-168=0$
$a=\pm 4.58$
$6{m}^{2}-5=0$
$0.03{y}^{2}=1.6$
$y=\pm 7.30$
$0.048{x}^{2}=2.01$
$1.001{x}^{2}-0.999=0$
$x=\pm 1.00$
(
[link] ) Graph the linear inequality
$3\left(x+2\right)<2\left(3x+4\right).$
( [link] ) Solve the fractional equation $\frac{x-1}{x+4}=\frac{x+3}{x-1}.$
$x=\frac{-11}{9}$
( [link] ) Find the product: $\sqrt{32{x}^{3}{y}^{5}}\sqrt{2{x}^{3}{y}^{3}}.$
( [link] ) Solve ${x}^{2}-4x=0.$
$x=0,\text{\hspace{0.17em}}4$
( [link] ) Solve ${y}^{2}-8y=-12.$
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