1. **State the problem:** Solve the quadratic equation $$0 = 24x + 2x^2 - 95$$ for $x$. 2. **Rewrite the equation in standard form:** $$2x^2 + 24x - 95 = 0$$ 3. **Identify coefficients:** - $a = 2$ - $b = 24$ - $c = -95$ 4. **Use the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 5. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 24^2 - 4 \times 2 \times (-95) = 576 + 760 = 1336$$ 6. **Find the square root of the discriminant:** $$\sqrt{1336} = \sqrt{4 \times 334} = 2\sqrt{334}$$ 7. **Substitute values into the quadratic formula:** $$x = \frac{-24 \pm 2\sqrt{334}}{2 \times 2} = \frac{-24 \pm 2\sqrt{334}}{4}$$ 8. **Simplify the expression:** $$x = \frac{-24}{4} \pm \frac{2\sqrt{334}}{4} = -6 \pm \frac{\sqrt{334}}{2}$$ 9. **Final solutions:** $$x_1 = -6 + \frac{\sqrt{334}}{2}$$ $$x_2 = -6 - \frac{\sqrt{334}}{2}$$ These are the two roots of the quadratic equation.