Solve Y=-0.5(x-50)^3+15

$$Y = - 0.5 ( x - 50 ) ^ { 3 } + 15$$

Solve for Y

$$Y=-\frac{\left(x-50\right)^{3}}{2}+15$$

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(x-50\right)^{3}.

$$Y=-0.5\left(x^{3}-150x^{2}+7500x-125000\right)+15$$

Use the distributive property to multiply -0.5 by x^{3}-150x^{2}+7500x-125000.

$$Y=-0.5x^{3}+75x^{2}-3750x+62500+15$$

Add 62500 and 15 to get 62515.

$$Y=-0.5x^{3}+75x^{2}-3750x+62515$$

Solve for x (complex solution)

$$x=\sqrt[3]{30-2Y}+50$$

$$x=e^{\frac{4i\pi }{3}}\sqrt[3]{30-2Y}+50$$

$$x=e^{\frac{2i\pi }{3}}\sqrt[3]{30-2Y}+50$$

Solve for x

$$x=\sqrt[3]{30-2Y}+50$$