1. **Problem statement:** Given points $P(6|6)$ and $Q(8|3)$ on a parabola $p$ with equation $y = -0.25x^2 + bx + c$, find $b$ and $c$. 2. **Formula and approach:** The parabola passes through $P$ and $Q$, so their coordinates satisfy the equation: $$y = -0.25x^2 + bx + c$$ Substitute $P(6,6)$ and $Q(8,3)$ to get two equations: $$6 = -0.25 \times 6^2 + 6b + c$$ $$3 = -0.25 \times 8^2 + 8b + c$$ 3. **Calculate values:** Calculate the squared terms: $$6^2 = 36, \quad 8^2 = 64$$ Substitute: $$6 = -0.25 \times 36 + 6b + c = -9 + 6b + c$$ $$3 = -0.25 \times 64 + 8b + c = -16 + 8b + c$$ Rewrite: $$6 = -9 + 6b + c \implies 6b + c = 15$$ $$3 = -16 + 8b + c \implies 8b + c = 19$$ 4. **Solve the system:** Subtract the first from the second: $$\cancel{8b} + c - (\cancel{6b} + c) = 19 - 15$$ $$2b = 4 \implies b = 2$$ Substitute $b=2$ into $6b + c = 15$: $$6 \times 2 + c = 15 \implies 12 + c = 15 \implies c = 3$$ 5. **Conclusion:** The parabola equation is: $$y = -0.25x^2 + 2x + 3$$ This confirms the given equation.