$$6x^{2}+x-3=0$$

$$x=\frac{-1±\sqrt{1^{2}-4\times 6\left(-3\right)}}{2\times 6}$$

$$x=\frac{-1±\sqrt{1-4\times 6\left(-3\right)}}{2\times 6}$$

$$x=\frac{-1±\sqrt{1-24\left(-3\right)}}{2\times 6}$$

$$x=\frac{-1±\sqrt{1+72}}{2\times 6}$$

$$x=\frac{-1±\sqrt{73}}{2\times 6}$$

$$x=\frac{-1±\sqrt{73}}{12}$$

$$x=\frac{\sqrt{73}-1}{12}$$

$$x=\frac{-\sqrt{73}-1}{12}$$

$$x=\frac{\sqrt{73}-1}{12}$$ $$x=\frac{-\sqrt{73}-1}{12}$$

$$6x^{2}+x-3=0$$

$$6x^{2}+x-3-\left(-3\right)=-\left(-3\right)$$

$$6x^{2}+x=-\left(-3\right)$$

$$6x^{2}+x=3$$

$$\frac{6x^{2}+x}{6}=\frac{3}{6}$$

$$x^{2}+\frac{1}{6}x=\frac{3}{6}$$

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

$$x^{2}+\frac{1}{6}x+\left(\frac{1}{12}\right)^{2}=\frac{1}{2}+\left(\frac{1}{12}\right)^{2}$$

$$x^{2}+\frac{1}{6}x+\frac{1}{144}=\frac{1}{2}+\frac{1}{144}$$

$$x^{2}+\frac{1}{6}x+\frac{1}{144}=\frac{73}{144}$$

$$\left(x+\frac{1}{12}\right)^{2}=\frac{73}{144}$$

$$\sqrt{\left(x+\frac{1}{12}\right)^{2}}=\sqrt{\frac{73}{144}}$$

$$x+\frac{1}{12}=\frac{\sqrt{73}}{12}$$ $$x+\frac{1}{12}=-\frac{\sqrt{73}}{12}$$

$$x=\frac{\sqrt{73}-1}{12}$$ $$x=\frac{-\sqrt{73}-1}{12}$$

$$x ^ 2 +\frac{1}{6}x -\frac{1}{2} = 0$$

$$r + s = -\frac{1}{6} $$ $$ rs = -\frac{1}{2}$$

$$r = -\frac{1}{12} - u$$ $$s = -\frac{1}{12} + u$$

$$(-\frac{1}{12} - u) (-\frac{1}{12} + u) = -\frac{1}{2}$$

$$\frac{1}{144} - u^2 = -\frac{1}{2}$$

$$-u^2 = -\frac{1}{2}-\frac{1}{144} = -\frac{73}{144}$$

$$u^2 = \frac{73}{144}$$ $$u = \pm\sqrt{\frac{73}{144}} = \pm \frac{\sqrt{73}}{12} $$

$$r =-\frac{1}{12} - \frac{\sqrt{73}}{12} = -0.795$$ $$s = -\frac{1}{12} + \frac{\sqrt{73}}{12} = 0.629$$