Answer
$\left\{\dfrac{3}{13}-\dfrac{2}{13}i,\dfrac{3}{13}+\dfrac{2}{13}i\right\}$
Work Step by Step
Subtract $6x$ from both sides.
$13x^2+1-6x=6x-6x$
Simplify by combining like terms.
$13x^2-6x+1=0$
The quadratic equation above has $a=13,b=-6$ and $c=1$.
The discriminant is
$b^2-4ac$
$=(-6)^2-4(13)(1)$
$=36-52$
$=-16$
Use Quadratic formula.
$x=\dfrac{-b\pm \sqrt {b^2-4ac}}{2a}$
Substitute the values of $a, b, c,$ and the discriminant:.
$x=\dfrac{-(-6)\pm \sqrt {-16}}{2(13)}$
Simplify.
$x=\dfrac{6\pm \sqrt{16(-1)}}{26}$
$x=\dfrac{6\pm 4i}{26}$
Factor out $2$.
$x=\dfrac{2(3\pm 2i)}{26}$
Cancel the common factor $2$.
$x=\dfrac{3\pm 2i}{13}$
The equation has two solutions: $\frac{3}{13}-\frac{2i}{13}$ and $\frac{3}{13}+\frac{2i}{13}$.
Check:
For $x=\frac{3}{13}-\frac{2i}{13}$.
$=13(\frac{3}{13}-\frac{2i}{13})^2-6(\frac{3}{13}-\frac{2i}{13})+1$
$=13(\frac{9}{169}-2\cdot \frac{6i}{169}+\frac{4i^2}{169})-\frac{18}{13}+\frac{12i}{13}+1$
$=\frac{9}{13}-\frac{12i}{13}+\frac{4i^2}{13}-\frac{18}{13}+\frac{12i}{13}+1$
$=\frac{4}{13}+\frac{4i^2}{13}$
$=\frac{4}{13}-\frac{4}{13}$
$=0$
For $x=\frac{3}{13}+\frac{2i}{13}$.
$=13(\frac{3}{13}+\frac{2i}{13})^2-6(\frac{3}{13}+\frac{2i}{13})+1$
$=13(\frac{9}{169}+2\cdot \frac{6i}{169}+\frac{4i^2}{169})-\frac{18}{13}-\frac{12i}{13}+1$
$=\frac{9}{13}+\frac{12i}{13}+\frac{4i^2}{13}-\frac{18}{13}-\frac{12i}{13}+1$
$=\frac{4}{13}+\frac{4i^2}{13}$
$=\frac{4}{13}-\frac{4}{13}$
$=0$
Hence, the solution set is $\left\{\dfrac{3}{13}-\dfrac{2}{13}i,\dfrac{3}{13}+\dfrac{2}{13}i\right\}$.
