# Chapter 11 - Additional Topics - 11.5 - Quadratic Equations: Complex Solutions - Problem Set 11.5: 19

{$\frac{1 - i\sqrt {2}}{3},\frac{1 + i\sqrt {2}}{3}$}

#### Work Step by Step

Step 1: Comparing $3x^{2}-2x+1=0$ to the standard form of a quadratic equation, $ax^{2}+bx+c=0$, we find: $a=3$, $b=-2$ and $c=1$ Step 2: The quadratic formula is: $x=\frac{-b \pm \sqrt {b^{2}-4ac}}{2a}$ Step 3: Substituting the values of a, b and c in the formula: $x=\frac{-(-2) \pm \sqrt {(-2)^{2}-4(3)(1)}}{2(3)}$ Step 4: $x=\frac{2 \pm \sqrt {4-12}}{6}$ Step 5: $x=\frac{2 \pm \sqrt {-8}}{6}$ Step 6: $x=\frac{2 \pm \sqrt {-1\times8}}{6}$ Step 7: $x=\frac{2 \pm (\sqrt {-1}\times\sqrt {4\times2})}{6}$ Step 8: $x=\frac{2 \pm (i\times 2\sqrt {2})}{6}$ Step 9: $x=\frac{1 \pm i\sqrt {2}}{3}$ Step 10: $x=\frac{1 - i\sqrt {2}}{3}$ or $x=\frac{1 + i\sqrt {2}}{3}$ Step 11: Therefore, the solution set is {$\frac{1 - i\sqrt {2}}{3},\frac{1 + i\sqrt {2}}{3}$}.

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