Question

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

Answer 1

We write, \[7x = 5x + 2x as\] as \[\sqrt{2} x^2 \times 5\sqrt{2} = 10 x^2 = 5x \times 2x\]

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

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

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

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

\[\Rightarrow x + \sqrt{2} = 0 \text { or } \sqrt{2}x + 5 = 0\]

\[ \Rightarrow x = - \sqrt{2} \text { or } x = - \frac{5}{\sqrt{2}} = - \frac{5\sqrt{2}}{2}\]Hence, the roots of the given equation are \[- \sqrt{2} \text { and} \] \[- \frac{5\sqrt{2}}{2}\].