1. **State the problem:** Simplify the expression $10147x^2 - 624.5x - 1$ or analyze it as a quadratic expression. 2. **Recall the quadratic form:** A quadratic expression is generally written as $ax^2 + bx + c$ where $a$, $b$, and $c$ are constants. 3. **Identify coefficients:** Here, $a = 10147$, $b = -624.5$, and $c = -1$. 4. **Discuss factorization or roots:** To factor or find roots, use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 5. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-624.5)^2 - 4 \times 10147 \times (-1)$$ $$= 390000.25 + 40588 = 430588.25$$ 6. **Calculate the roots:** $$x = \frac{624.5 \pm \sqrt{430588.25}}{2 \times 10147}$$ $$\sqrt{430588.25} \approx 656.3$$ 7. **Evaluate each root:** $$x_1 = \frac{624.5 + 656.3}{20294} = \frac{1280.8}{20294} \approx 0.0631$$ $$x_2 = \frac{624.5 - 656.3}{20294} = \frac{-31.8}{20294} \approx -0.00157$$ 8. **Conclusion:** The quadratic $10147x^2 - 624.5x - 1$ has roots approximately $x \approx 0.0631$ and $x \approx -0.00157$. This completes the analysis of the quadratic expression.