Solve R=-3(d-2)^2+12

$$R = - 3 ( d - 2 ) ^ { 2 } + 12$$

Solve for R

$$R=3d\left(4-d\right)$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(d-2\right)^{2}.

$$R=-3\left(d^{2}-4d+4\right)+12$$

Use the distributive property to multiply -3 by d^{2}-4d+4.

$$R=-3d^{2}+12d-12+12$$

Add -12 and 12 to get 0.

$$R=-3d^{2}+12d$$

Solve for d

$$d=-\frac{\sqrt{36-3R}}{3}+2$$

$$d=\frac{\sqrt{36-3R}}{3}+2\text{, }R\leq 12$$