Solve the following quadratic equations:\(x^2 -(\sqrt{2}+i)x + \sqrt{2

1.	Solve the following quadratic equations:\(x^2 -(\sqrt{2}+i)x + \sqrt{2}i = 0\)
Answer» \(x^2 -(\sqrt{2}+i)x + \sqrt{2}i = 0\) Given \(x^2 -(\sqrt{2}+i)x + \sqrt{2}i = 0\) ⇒ \(x^2 -(\sqrt{2}x+ix) + \sqrt{2}i = 0\) ⇒ \(x^2 -\sqrt{2}x-ix + \sqrt{2}i = 0\) ⇒ \(x(x-\sqrt{2})-i(x-\sqrt{2})=0\) ⇒\((x-\sqrt{2})(x-i)=0\) ⇒ \(x-\sqrt{2}=0\,or\, x=i=0\) ∴ x = \(\sqrt{2}\,or\,i\) Thus, the roots of the given equation are \(\sqrt{2}\) and i.

Solve the following quadratic equations:\(x^2 -(\sqrt{2}+i)x + \sqrt{2}i = 0\)

Answer»

\(x^2 -(\sqrt{2}+i)x + \sqrt{2}i = 0\)

Given \(x^2 -(\sqrt{2}+i)x + \sqrt{2}i = 0\)

⇒ \(x^2 -(\sqrt{2}x+ix) + \sqrt{2}i = 0\)

⇒ \(x^2 -\sqrt{2}x-ix + \sqrt{2}i = 0\)

⇒ \(x(x-\sqrt{2})-i(x-\sqrt{2})=0\)

⇒\((x-\sqrt{2})(x-i)=0\)

⇒ \(x-\sqrt{2}=0\,or\, x=i=0\)

∴ x = \(\sqrt{2}\,or\,i\)

Thus, the roots of the given equation are \(\sqrt{2}\) and i.