Discuss the nature of roots of the following equations: (i)

    \[\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{7x}\text{ }+\text{ }\mathbf{8}\text{ }=\text{ }\mathbf{0}\]

(ii)

    \[{{\mathbf{x}}^{\mathbf{2}}}~\text{ }{\scriptscriptstyle 1\!/\!{ }_2}\text{ }\mathbf{x}\text{ }\text{ }\mathbf{4}\text{ }=\text{ }\mathbf{0}\]

Discuss the nature of roots of the following equations: (i)

    \[\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{7x}\text{ }+\text{ }\mathbf{8}\text{ }=\text{ }\mathbf{0}\]

(ii)

    \[{{\mathbf{x}}^{\mathbf{2}}}~\text{ }{\scriptscriptstyle 1\!/\!{ }_2}\text{ }\mathbf{x}\text{ }\text{ }\mathbf{4}\text{ }=\text{ }\mathbf{0}\]

(i)

    \[\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{7x}\text{ }+\text{ }\mathbf{8}\text{ }=\text{ }\mathbf{0}\]

Let us consider,

    \[a\text{ }=\text{ }3,\text{ }b\text{ }=\text{ }-7,\text{ }c\text{ }=\text{ }8\]

By using the formula,

    \[\begin{array}{*{35}{l}} D\text{ }=\text{ }{{b}^{2}}~\text{ }4ac  \\ =\text{ }{{\left( -7 \right)}^{2}}~\text{ }4\left( 3 \right)\text{ }\left( 8 \right)  \\ =\text{ }49\text{ }\text{ }96  \\ =\text{ }-47  \\ \end{array}\]

So,

Discriminate,

    \[\begin{array}{*{35}{l}} D\text{ }=\text{ }-47  \\ D\text{ }<\text{ }0  \\ \end{array}\]

∴ Roots are not real.

(ii)

    \[{{x}^{2}}~\text{ }{\scriptscriptstyle 1\!/\!{ }_2}\text{ }x\text{ }\text{ }4\text{ }=\text{ }0\]

Let us consider,

    \[a\text{ }=\text{ }1,\text{ }b\text{ }=\text{ }-1/2,\text{ }c\text{ }=\text{ }-4\]

By using the formula,

    \[\begin{array}{*{35}{l}} D\text{ }=\text{ }{{b}^{2}}~\text{ }4ac  \\ =\text{ }{{\left( -1/2 \right)}^{2}}~\text{ }4\left( 1 \right)\text{ }\left( -4 \right)  \\ =\text{ }1/4\text{ }+\text{ }16  \\ =\text{ }65/16  \\ \end{array}\]

So,

Discriminate,

    \[\begin{array}{*{35}{l}} D\text{ }=\text{ }65/16  \\ D\text{ }>\text{ }0  \\ \end{array}\]

∴ Roots are real and distinct.

So,

    \[\begin{array}{*{35}{l}} x\text{ }=\text{ }\left[ -\left( -1/2 \right)~\pm \text{ }\surd (65/16 \right)]\text{ }/\text{ }2\left( 1 \right)  \\ =\text{ }\left[ 1/2~\pm \text{ }\surd 65/4 \right]\text{ }/\text{ }2  \\ =\text{ }\left[ 1/2\text{ }+~\surd 65/4 \right]/\text{ }2\text{ }or\text{ }\left[ 1/2\text{ }~\surd 65/4 \right]/\text{ }2  \\ =\text{ }\left( 2\text{ }+~\surd 65 \right)/4\text{ }/2\text{ }or\text{ }\left( 2\text{ }~\surd 65 \right)/4\text{ }/2  \\ =\text{ }\left( 2\text{ }+~\surd 65 \right)/\text{ }8\text{ }or\text{ }\left( 2\text{ }~\surd 65 \right)/\text{ }8  \\ \end{array}\]

∴ Value of x =

    \[\left( 2\text{ }+~\surd 65 \right)/\text{ }8,\text{ }\left( 2\text{ }~\surd 65 \right)/\text{ }8\]