<< Chapter < Page | Chapter >> Page > |
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.
Solve $4{n}^{2}=24{m}^{2}{p}^{8}$ for $n.$
$n=\pm m{p}^{4}\sqrt{6}$
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.
For the following problems, solve each of the quadratic equations using the method of extraction of roots.
${x}^{2}=49$
${a}^{2}=4$
${a}^{2}=1$
${x}^{2}=81$
${a}^{2}=10$
${b}^{2}=6$
${y}^{2}=7$
${a}^{2}-3=0$
${y}^{2}-1=0$
${x}^{2}-11=0$
$5{b}^{2}-5=0$
$4{a}^{2}=40$
For the following problems, solve for the indicated variable.
${x}^{2}=4{a}^{2},$ for $x$
${a}^{2}=25{c}^{2},$ for $a$
${k}^{2}={p}^{2}{q}^{2}{r}^{2},$ for $k$
$9{y}^{2}=27{x}^{2}{z}^{4},$ for $y$
${x}^{2}-{z}^{2}=0,$ for $z$
For the following problems, solve each of the quadratic equations using the method of extraction of roots.
${\left(x-1\right)}^{2}=4$
${\left(x-3\right)}^{2}=25$
${\left(a+3\right)}^{2}=49$
${\left(a-6\right)}^{2}=3$
${\left(x+4\right)}^{2}=5$
$a=-4\text{\hspace{0.17em}}\pm \sqrt{5}$
${\left(b+6\right)}^{2}=7$
${\left(x+1\right)}^{2}=a,$ for $x$
$x=-1\text{\hspace{0.17em}}\pm \sqrt{a}$
${\left(y+5\right)}^{2}=b,$ for $y$
${\left(x+10\right)}^{2}={c}^{2},$ for $x$
${\left(x+c\right)}^{2}={a}^{2},$ for $x$
For the following problems, round each result to the nearest hundredth.
$6{m}^{2}-5=0$
$0.048{x}^{2}=2.01$
(
[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 ${y}^{2}-8y=-12.$
Notification Switch
Would you like to follow the 'Elementary algebra' conversation and receive update notifications?