1. **State the problem:** Solve the quadratic equation $10147x^2 - 624.5x - 1 = 0$ for $x$. 2. **Formula used:** The quadratic formula is given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a$, $b$, and $c$ are coefficients from the quadratic equation $ax^2 + bx + c = 0$. 3. **Identify coefficients:** Here, $a = 10147$, $b = -624.5$, and $c = -1$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-624.5)^2 - 4 \times 10147 \times (-1)$$ $$= 390000.25 + 40588 = 430588.25$$ 5. **Calculate the square root of the discriminant:** $$\sqrt{430588.25} \approx 656.3$$ 6. **Apply the quadratic formula:** $$x = \frac{-(-624.5) \pm 656.3}{2 \times 10147} = \frac{624.5 \pm 656.3}{20294}$$ 7. **Calculate the two roots:** - For the plus sign: $$x_1 = \frac{624.5 + 656.3}{20294} = \frac{1280.8}{20294} \approx 0.0631$$ - For the minus sign: $$x_2 = \frac{624.5 - 656.3}{20294} = \frac{-31.8}{20294} \approx -0.00157$$ **Final answer:** The solutions to the equation are approximately $$x \approx 0.0631 \text{ and } x \approx -0.00157$$