1. The problem asks us to express the quadratic expression $$q + 12x - qx^2$$ in the form $$a - b(x - c)^2$$ where $$a$$, $$b$$, and $$c$$ are in terms of $$q$$. 2. Start with the given expression: $$q + 12x - qx^2$$ Rewrite it in standard quadratic form: $$-qx^2 + 12x + q$$ 3. To write it as $$a - b(x - c)^2$$, recognize this is a vertex form of a parabola. The vertex form is: $$a - b(x - c)^2 = a - b(x^2 - 2cx + c^2) = a - b x^2 + 2bc x - b c^2$$ 4. Match coefficients from the original expression and the vertex form: - Coefficient of $$x^2$$: $$-q = -b$$ so $$b = q$$ - Coefficient of $$x$$: $$12 = 2 b c = 2 q c$$ so $$c = \frac{12}{2 q} = \frac{6}{q}$$ - Constant term: $$q = a - b c^2 = a - q \left(\frac{6}{q}\right)^2 = a - q \frac{36}{q^2} = a - \frac{36}{q}$$ 5. Solve for $$a$$: $$a = q + \frac{36}{q}$$ 6. Final expressions: $$a = q + \frac{36}{q}$$ $$b = q$$ $$c = \frac{6}{q}$$ These express $$a$$, $$b$$, and $$c$$ in terms of $$q$$ so that $$q + 12x - qx^2 = a - b(x - c)^2$$.