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