Solve the equations:

(a) $3x^{2}-5x+1=0$

(b) $x^{2}+x=-1$

Solution (a) The discriminant: $b^{2}-4ac=25-12=13$. We use the quadratic formula. $$x=\dfrac{-b\pm \displaystyle\sqrt{b^{2}-4ac}}{2a}=\dfrac{5\pm \displaystyle\sqrt{13}}{6}$$

The solutions are $x=\dfrac{5-\sqrt{13}}{6}\approx 0.232$ and $x=\dfrac{5+ \sqrt{13}}{6}\approx 1.434$

(b) The equation in standard form is $x^2+x+1=0$. Its discriminant is $b^{2}-4ac=1-4=-3$. This equation has no real solution.