1. The problem is to find the value of $a$ such that the line $y = -1.5$ intersects the parabola $y = x^2 + 8x + a$. 2. Set the two expressions for $y$ equal to each other to find the intersection points: $$-1.5 = x^2 + 8x + a$$ 3. Rearrange the equation to standard quadratic form: $$x^2 + 8x + a + 1.5 = 0$$ 4. For the line to intersect the parabola, this quadratic must have real solutions. The discriminant $\Delta$ must be greater than or equal to zero: $$\Delta = b^2 - 4ac = 8^2 - 4 \times 1 \times (a + 1.5) \geq 0$$ 5. Calculate the discriminant inequality: $$64 - 4(a + 1.5) \geq 0$$ 6. Simplify: $$64 - 4a - 6 \geq 0$$ $$58 - 4a \geq 0$$ 7. Solve for $a$: $$-4a \geq -58$$ $$\cancel{-4}a \leq \cancel{-58} \quad \Rightarrow \quad a \leq \frac{58}{4}$$ 8. Simplify the fraction: $$a \leq 14.5$$ **Final answer:** The value of $a$ must satisfy $a \leq 14.5$ for the line $y = -1.5$ to intersect the parabola $y = x^2 + 8x + a$.