Solve (3sqrt{3}-(2sqrt{3}-sqrt{3}*(x-2))*0.5)*(8-x)*0.5

$$(3 \sqrt{ 3 } -(2 \sqrt{ 3 } - \sqrt{ 3 } \times (x-2)) \times 0.5) \times (8-x) \times 0.5$$

Evaluate

$$\frac{\sqrt{3}\left(8-x\right)\left(\frac{x}{2}+1\right)}{2}$$

Solution Steps

Use the distributive property to multiply \sqrt{3} by x-2.

$$0.5\left(3\sqrt{3}-0.5\left(2\sqrt{3}-\left(\sqrt{3}x-2\sqrt{3}\right)\right)\right)\left(8-x\right)$$

To find the opposite of \sqrt{3}x-2\sqrt{3}, find the opposite of each term.

$$0.5\left(3\sqrt{3}-0.5\left(2\sqrt{3}-\sqrt{3}x-\left(-2\sqrt{3}\right)\right)\right)\left(8-x\right)$$

The opposite of -2\sqrt{3} is 2\sqrt{3}.

$$0.5\left(3\sqrt{3}-0.5\left(2\sqrt{3}-\sqrt{3}x+2\sqrt{3}\right)\right)\left(8-x\right)$$

Combine 2\sqrt{3} and 2\sqrt{3} to get 4\sqrt{3}.

$$0.5\left(3\sqrt{3}-0.5\left(4\sqrt{3}-\sqrt{3}x\right)\right)\left(8-x\right)$$

Use the distributive property to multiply -0.5 by 4\sqrt{3}-\sqrt{3}x.

$$0.5\left(3\sqrt{3}-2\sqrt{3}+0.5\sqrt{3}x\right)\left(8-x\right)$$

Combine 3\sqrt{3} and -2\sqrt{3} to get \sqrt{3}.

$$0.5\left(\sqrt{3}+0.5\sqrt{3}x\right)\left(8-x\right)$$

Use the distributive property to multiply 0.5 by \sqrt{3}+0.5\sqrt{3}x.

$$\left(0.5\sqrt{3}+0.25\sqrt{3}x\right)\left(8-x\right)$$

Apply the distributive property by multiplying each term of 0.5\sqrt{3}+0.25\sqrt{3}x by each term of 8-x.

$$4\sqrt{3}-0.5\sqrt{3}x+2\sqrt{3}x-0.25\sqrt{3}x^{2}$$

Combine -0.5\sqrt{3}x and 2\sqrt{3}x to get 1.5\sqrt{3}x.

$$4\sqrt{3}+1.5\sqrt{3}x-0.25\sqrt{3}x^{2}$$

Differentiate w.r.t. x

$$\sqrt{3}\times \frac{3-x}{2}$$