Answer: [tex]x_1=\frac{1}{5}+\frac{\sqrt{29}}{5}i\\x_2=\frac{1}{5}-\frac{\sqrt{29}}{5}i[/tex]
Step-by-step explanation:
1. You can solve the quadratic equation by completing the square, as following:
- The leading coefficient must be 1, therefore, you need to divide the equation by 5:
[tex]\frac{5x^{2}-2x+6}{5}=0\\x^{2}-\frac{2}{5}x+\frac{6}{5}=0[/tex]
- Substract [tex]\frac{6}{5}[/tex] at both sides:
[tex]x^{2}-\frac{2}{5}x+\frac{6}{5}-\frac{6}{5}=-\frac{6}{5}[/tex]
[tex]x^{2}-\frac{2}{5}x=-\frac{6}{5}[/tex]
- Divide the coefficient of [tex]x[/tex] by 2 and and square it:
[tex](\frac{2}{\frac{5}{2}})^{2} =\frac{1}{25}[/tex]
- Add it to both sides:
[tex]x^{2}-\frac{2}{5}x-\frac{1}{25}=-\frac{6}{5}+\frac{1}{25}[/tex]
- Then:
[tex](x-\frac{1}{5})^{2} =-\frac{29}{25}[/tex]
[tex]x-\frac{1}{5}=\sqrt{-\frac{29}{25}}[/tex]
- Knowing that [tex]i=\sqrt{-1}[/tex]:
[tex]x_1=\frac{1}{5}+\frac{\sqrt{29}}{5}i\\x_2=\frac{1}{5}-\frac{\sqrt{29}}{5}i[/tex]
