1. The problem is to solve the quadratic equation $7.5x^2 + 4x - 8 = 0$ for $x$. 2. We use the quadratic formula to solve equations of the form $ax^2 + bx + c = 0$: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a = 7.5$, $b = 4$, and $c = -8$. 3. Calculate the discriminant: $$\Delta = b^2 - 4ac = 4^2 - 4 \times 7.5 \times (-8) = 16 + 240 = 256$$ 4. Since $\Delta > 0$, there are two real solutions. 5. Substitute values into the quadratic formula: $$x = \frac{-4 \pm \sqrt{256}}{2 \times 7.5} = \frac{-4 \pm 16}{15}$$ 6. Calculate each solution: - For the plus sign: $$x = \frac{-4 + 16}{15} = \frac{12}{15} = \frac{\cancel{12}}{\cancel{15}} = \frac{4}{5} = 0.8$$ - For the minus sign: $$x = \frac{-4 - 16}{15} = \frac{-20}{15} = \frac{\cancel{-20}}{\cancel{15}} = \frac{-4}{3} \approx -1.333$$ 7. Final answer: $$x = 0.8 \quad \text{or} \quad x = -1.333$$