1.
[tex]x^{2}+\sqrt{2}x - 4 = 0\\\\a = 1, \ b = \sqrt{2}, \ c = -4\\\\\Delta = \sqrt{2}^{2}-4\cdot1\cdot(-4) = 2+16 = 18\\\\\sqrt{\Delta} = \sqrt{18} =\sqrt{9\cdot2} = 3\sqrt{2}\\\\x_1 = \frac{-\sqrt{2}-3\sqrt{2}}{2} = \frac{-4\sqrt{2}}{2} = -2\sqrt{2}\\\\x_2 = \frac{-\sqrt{2}+3\sqrt{2}}{2} = \frac{2\sqrt{2}}{2} = \sqrt{2}\\\\\boxed{x \in \{-2\sqrt{2}, \sqrt{2}\}}[/tex]
2.
[tex](x^{2}-6)(x^{2}-7x+6) = 0\\\\(x+\sqrt{6})(x-\sqrt{6})(x^{2}-6x-x+6) = 0\\\\(x+\sqrt{6})(x-\sqrt{6})[x(x-6)-(x-6)] = 0\\\\(x+\sqrt{6})(x-\sqrt{6})(x-6)(x-1) = 0\\\\x+\sqrt{6} = 0 \ \vee \ x-\sqrt{6}=0 \ \vee \ x-6 =0 \ \vee \ x = 1\\\\x = -\sqrt{6} \ \vee \ x = \sqrt{6} \ \vee \ x = 6 \ \vee \ x = 1\\\\\boxed{x\in\{-\sqrt{6}, 1,\sqrt{6},6\}}[/tex]
3.
[tex]x^{2}(x-6)-2x(x-6) = 0\\\\(x-6)(x^{2}-2x) = 0\\\\(x-6)\cdot x\cdot(x-2) = 0\\\\x(x-6)(x-2) = 0\\\\x = 0 \ \vee \ x-6 = 0 \ \vee \ x-2 = 0\\\\x = 0 \ \vee \ x = 6 \ \vee \ x = 2\\\\\boxed{x\in \{0, 2, 6\}}[/tex]
