1. **State the problem:** Find the value(s) of $x$ when $-2x^2 - 5x + 7 = 0$. 2. **Formula used:** This is a quadratic equation of the form $ax^2 + bx + c = 0$. The solutions are given by the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a = -2$, $b = -5$, and $c = 7$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-5)^2 - 4(-2)(7) = 25 + 56 = 81$$ Since $\Delta > 0$, there are two real solutions. 4. **Apply the quadratic formula:** $$x = \frac{-(-5) \pm \sqrt{81}}{2(-2)} = \frac{5 \pm 9}{-4}$$ 5. **Find the two solutions:** - For $+$ sign: $$x = \frac{5 + 9}{-4} = \frac{14}{-4} = -3.5$$ - For $-$ sign: $$x = \frac{5 - 9}{-4} = \frac{-4}{-4} = 1$$ **Final answer:** The values of $x$ are $-3.5$ and $1$.