1. **State the problem:** We have the system of equations: $$y = -1.5$$ $$y = x^2 + 8x + a$$ where $a$ is a positive constant. We want to find the value of $a$ such that the system has exactly one distinct real solution. 2. **Understand the condition for one distinct real solution:** The line intersects the parabola at exactly one point, meaning the quadratic equation formed by setting the two expressions for $y$ equal has exactly one solution. This happens when the discriminant is zero. 3. **Set the equations equal:** $$-1.5 = x^2 + 8x + a$$ Rearranged: $$x^2 + 8x + a + 1.5 = 0$$ 4. **Use the quadratic formula discriminant:** For a quadratic $ax^2 + bx + c = 0$, the discriminant is: $$\Delta = b^2 - 4ac$$ Here, $a=1$, $b=8$, and $c = a + 1.5$ (note the variable $a$ in the problem is the constant term, not the quadratic coefficient). 5. **Set discriminant to zero for one solution:** $$0 = 8^2 - 4 \times 1 \times (a + 1.5)$$ $$0 = 64 - 4(a + 1.5)$$ 6. **Solve for $a$:** $$4(a + 1.5) = 64$$ $$\cancel{4}(a + 1.5) = \cancel{4}16$$ $$a + 1.5 = 16$$ $$a = 16 - 1.5$$ $$a = 14.5$$ 7. **Check positivity:** $a = 14.5$ is positive, which satisfies the problem condition. **Final answer:** $$\boxed{14.5}$$