question_answer

Find the roots of the quadratic equation \[\sqrt{2}{{x}^{2}}+7x+5\sqrt{2}=0\].

Answer:

Given quadratic equation is,

\[\sqrt{2}{{x}^{2}}+7x+5\sqrt{2}=0\]

\[\sqrt{2}{{x}^{2}}+5x+2x+5\sqrt{2}=0\]

[Splitting middle term]

\[\Rightarrow \]   \[x\left( \sqrt{2}x+5 \right)+\sqrt{2}\left( \sqrt{2}+5 \right)=0\]

\[\Rightarrow \]              \[\left( \sqrt{2}x+5 \right)\left( x+\sqrt{2} \right)=0\]

\[\Rightarrow \] either \[\left( \sqrt{2}x+5 \right)=0\] or \[\left( x+\sqrt{2} \right)=0\]

\[\Rightarrow \]                   \[x=-\frac{5}{\sqrt{2}}\] or \[x=-\sqrt{2}\]

Hence roots are \[\frac{-5}{\sqrt{2}}\] and \[-\sqrt{2}\]