(a) Coefficients a = 2 a=\sqrt2 a = 2 b = 1 b=1 b = 1 c = − 2 c=-\sqrt2 c = − 2 Δ = b 2 − 4 a c = 1 2 − 4 ⋅ 2 ⋅ ( − 2 ) = 1 + 4 ⋅ 2 = 9. \Delta=b^2-4ac=1^2-4\cdot\sqrt2\cdot(-\sqrt2)=1+4\cdot2=9. Δ = b 2 − 4 a c = 1 2 − 4 ⋅ 2 ⋅ ( − 2 ) = 1 + 4 ⋅ 2 = 9. x = − 1 ± 9 2 2 = − 1 ± 3 2 2 . x=\frac{-1\pm\sqrt9}{2\sqrt2}=\frac{-1\pm3}{2\sqrt2}. x = 2 2 − 1 ± 9 = 2 2 − 1 ± 3 . x 1 = 2 2 2 = 1 2 = 2 2 ≈ 0.707 , x 2 = − 4 2 2 = − 2 2 = − 2 ≈ − 1.414. x_1=\frac{2}{2\sqrt2}=\frac{1}{\sqrt2}=\frac{\sqrt2}{2}\approx0.707,\qquad x_2=\frac{-4}{2\sqrt2}=\frac{-2}{\sqrt2}=-\sqrt2\approx-1.414. x 1 = 2 2 2 = 2 1 = 2 2 ≈ 0.707 , x 2 = 2 2 − 4 = 2 − 2 = − 2 ≈ − 1.414.
(b) Coefficients a = 1 a=1 a = 1 b = 6 − 2 b=\sqrt6-\sqrt2 b = 6 − 2 c = − 2 3 c=-2\sqrt3 c = − 2 3 Δ = ( 6 − 2 ) 2 − 4 ⋅ 1 ⋅ ( − 2 3 ) = ( 6 − 2 12 + 2 ) + 8 3 = 8 − 4 3 + 8 3 = 8 + 4 3 . \Delta=(\sqrt6-\sqrt2)^2-4\cdot1\cdot(-2\sqrt3)=(6-2\sqrt{12}+2)+8\sqrt3=8-4\sqrt3+8\sqrt3=8+4\sqrt3. Δ = ( 6 − 2 ) 2 − 4 ⋅ 1 ⋅ ( − 2 3 ) = ( 6 − 2 12 + 2 ) + 8 3 = 8 − 4 3 + 8 3 = 8 + 4 3 . ( 6 + 2 ) 2 = 6 + 2 12 + 2 = 8 + 4 3 (\sqrt6+\sqrt2)^2=6+2\sqrt{12}+2=8+4\sqrt3 ( 6 + 2 ) 2 = 6 + 2 12 + 2 = 8 + 4 3 Δ = 6 + 2 \sqrt{\Delta}=\sqrt6+\sqrt2 Δ = 6 + 2 x = − ( 6 − 2 ) ± ( 6 + 2 ) 2 . x=\frac{-(\sqrt6-\sqrt2)\pm(\sqrt6+\sqrt2)}{2}. x = 2 − ( 6 − 2 ) ± ( 6 + 2 ) . + + + x 1 = − 6 + 2 + 6 + 2 2 = 2 2 2 = 2 ≈ 1.414. x_1=\frac{-\sqrt6+\sqrt2+\sqrt6+\sqrt2}{2}=\frac{2\sqrt2}{2}=\sqrt2\approx1.414. x 1 = 2 − 6 + 2 + 6 + 2 = 2 2 2 = 2 ≈ 1.414. − - − x 2 = − 6 + 2 − 6 − 2 2 = − 2 6 2 = − 6 ≈ − 2.449. x_2=\frac{-\sqrt6+\sqrt2-\sqrt6-\sqrt2}{2}=\frac{-2\sqrt6}{2}=-\sqrt6\approx-2.449. x 2 = 2 − 6 + 2 − 6 − 2 = 2 − 2 6 = − 6 ≈ − 2.449. x a = 2 2 , − 2 x b = 2 , − 6 \boxed{x_a=\frac{\sqrt2}{2},\,-\sqrt2 \qquad x_b=\sqrt2,\,-\sqrt6} x a = 2 2 , − 2 x b = 2 , − 6