![\[{{\mathbf{x}}^{\mathbf{2}}}~\text{ }{\scriptscriptstyle 1\!/\!{ }_2}\text{ }\mathbf{x}\text{ }\text{ }\mathbf{4}\text{ }=\text{ }\mathbf{0}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-2c22ed38fc4cb9fa07c48cc9f20ebc0b_l3.png "Rendered by QuickLaTeX.com")
(i)
![\[\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{7x}\text{ }+\text{ }\mathbf{8}\text{ }=\text{ }\mathbf{0}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-283b43bb6a5c3d0da5e744443eeb8bdc_l3.png "Rendered by QuickLaTeX.com")
Let us consider,
![\[a\text{ }=\text{ }3,\text{ }b\text{ }=\text{ }-7,\text{ }c\text{ }=\text{ }8\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-d56f769cc3183ae7e6585ee111fe0079_l3.png "Rendered by QuickLaTeX.com")
By using the formula,
![\[\begin{array}{*{35}{l}} D\text{ }=\text{ }{{b}^{2}}~\text{ }4ac \\ =\text{ }{{\left( -7 \right)}^{2}}~\text{ }4\left( 3 \right)\text{ }\left( 8 \right) \\ =\text{ }49\text{ }\text{ }96 \\ =\text{ }-47 \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-8d61bf760ab569108f66888f6e218bfb_l3.png "Rendered by QuickLaTeX.com")
So,
Discriminate,
![\[\begin{array}{*{35}{l}} D\text{ }=\text{ }-47 \\ D\text{ }<\text{ }0 \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-88ce2eb0b2dbd9bfc406f6f57b0fec0c_l3.png "Rendered by QuickLaTeX.com")
∴ Roots are not real.
(ii)
![\[{{x}^{2}}~\text{ }{\scriptscriptstyle 1\!/\!{ }_2}\text{ }x\text{ }\text{ }4\text{ }=\text{ }0\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-db0d540b427e162bc2ffc43283c38cc9_l3.png "Rendered by QuickLaTeX.com")
Let us consider,
![\[a\text{ }=\text{ }1,\text{ }b\text{ }=\text{ }-1/2,\text{ }c\text{ }=\text{ }-4\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-6db480ce05ee8502205b6876fa573de2_l3.png "Rendered by QuickLaTeX.com")
By using the formula,
![\[\begin{array}{*{35}{l}} D\text{ }=\text{ }{{b}^{2}}~\text{ }4ac \\ =\text{ }{{\left( -1/2 \right)}^{2}}~\text{ }4\left( 1 \right)\text{ }\left( -4 \right) \\ =\text{ }1/4\text{ }+\text{ }16 \\ =\text{ }65/16 \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-9d0fd85ce3c489f5f3233026e710a590_l3.png "Rendered by QuickLaTeX.com")
So,
Discriminate,
![\[\begin{array}{*{35}{l}} D\text{ }=\text{ }65/16 \\ D\text{ }>\text{ }0 \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-bef004159069617485cb5a8b0942edc3_l3.png "Rendered by QuickLaTeX.com")
∴ Roots are real and distinct.
So,
![\[\begin{array}{*{35}{l}} x\text{ }=\text{ }\left[ -\left( -1/2 \right)~\pm \text{ }\surd (65/16 \right)]\text{ }/\text{ }2\left( 1 \right) \\ =\text{ }\left[ 1/2~\pm \text{ }\surd 65/4 \right]\text{ }/\text{ }2 \\ =\text{ }\left[ 1/2\text{ }+~\surd 65/4 \right]/\text{ }2\text{ }or\text{ }\left[ 1/2\text{ }~\surd 65/4 \right]/\text{ }2 \\ =\text{ }\left( 2\text{ }+~\surd 65 \right)/4\text{ }/2\text{ }or\text{ }\left( 2\text{ }~\surd 65 \right)/4\text{ }/2 \\ =\text{ }\left( 2\text{ }+~\surd 65 \right)/\text{ }8\text{ }or\text{ }\left( 2\text{ }~\surd 65 \right)/\text{ }8 \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-524ee5f0d9cd2868c7188b2fafc5000b_l3.png "Rendered by QuickLaTeX.com")
∴ Value of x =
![\[\left( 2\text{ }+~\surd 65 \right)/\text{ }8,\text{ }\left( 2\text{ }~\surd 65 \right)/\text{ }8\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-647c7e164bdda1e9311d61e51f191094_l3.png "Rendered by QuickLaTeX.com")
