Question

Solve each of the quadratic equations 0=5x^2-2x+6

Answer 1

Answer: $x_1=\frac{1}{5}+\frac{\sqrt{29}}{5}i\\x_2=\frac{1}{5}-\frac{\sqrt{29}}{5}i$

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:

$\frac{5x^{2}-2x+6}{5}=0\\x^{2}-\frac{2}{5}x+\frac{6}{5}=0$

- Substract $\frac{6}{5}$ at both sides:

$x^{2}-\frac{2}{5}x+\frac{6}{5}-\frac{6}{5}=-\frac{6}{5}$

$x^{2}-\frac{2}{5}x=-\frac{6}{5}$

- Divide the coefficient of $x$ by 2 and and square it:

$(\frac{2}{\frac{5}{2}})^{2} =\frac{1}{25}$

- Add it to both sides:

$x^{2}-\frac{2}{5}x-\frac{1}{25}=-\frac{6}{5}+\frac{1}{25}$

- Then:

$(x-\frac{1}{5})^{2} =-\frac{29}{25}$

$x-\frac{1}{5}=\sqrt{-\frac{29}{25}}$

- Knowing that $i=\sqrt{-1}$:

$x_1=\frac{1}{5}+\frac{\sqrt{29}}{5}i\\x_2=\frac{1}{5}-\frac{\sqrt{29}}{5}i$