Quadratic Solve F46153

1. **State the problem:** Solve the quadratic equation $$7x^2 + x - 58 = 0$$ for $x$, rounding to the nearest thousandth. 2. **Formula used:** The quadratic formula is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=7$, $b=1$, and $c=-58$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 1^2 - 4 \times 7 \times (-58) = 1 + 1624 = 1625$$ 4. **Apply the quadratic formula:** $$x = \frac{-1 \pm \sqrt{1625}}{2 \times 7} = \frac{-1 \pm \sqrt{1625}}{14}$$ 5. **Simplify the square root:** $$\sqrt{1625} \approx 40.311$$ 6. **Find the two solutions:** $$x_1 = \frac{-1 + 40.311}{14} = \frac{39.311}{14}$$ $$x_2 = \frac{-1 - 40.311}{14} = \frac{-41.311}{14}$$ 7. **Simplify fractions with cancellation:** $$x_1 = \frac{\cancel{39.311}}{\cancel{14}} \approx 2.808$$ $$x_2 = \frac{\cancel{-41.311}}{\cancel{14}} \approx -2.951$$ 8. **Order solutions from least to greatest:** $$x = -2.951, 2.808$$ **Final answer:** $$x = -2.951, 2.808$$