Solve 1/2left(6-tright)left(6sqrt{3}-sqrt{3}tright)+6/2left(6sqrt{3}-6sqrt{3}+sqrt{3}tright)

$$\frac{ 1 }{ 2 } \left( 6-t \right) \left( 6 \sqrt{ 3 } - \sqrt{ 3 } t \right) + \frac{ 6 }{ 2 } \left( 6 \sqrt{ 3 } -6 \sqrt{ 3 } + \sqrt{ 3 } t \right)$$

Evaluate

$$\frac{\sqrt{3}\left(t^{2}-6t+36\right)}{2}$$

Solution Steps

Use the distributive property to multiply \frac{1}{2} by 6-t.

$$\left(\frac{1}{2}\times 6+\frac{1}{2}\left(-1\right)t\right)\left(6\sqrt{3}-\sqrt{3}t\right)+\frac{6}{2}\left(6\sqrt{3}-6\sqrt{3}+\sqrt{3}t\right)$$

Multiply \frac{1}{2} and 6 to get \frac{6}{2}.

$$\left(\frac{6}{2}+\frac{1}{2}\left(-1\right)t\right)\left(6\sqrt{3}-\sqrt{3}t\right)+\frac{6}{2}\left(6\sqrt{3}-6\sqrt{3}+\sqrt{3}t\right)$$

Divide 6 by 2 to get 3.

$$\left(3+\frac{1}{2}\left(-1\right)t\right)\left(6\sqrt{3}-\sqrt{3}t\right)+\frac{6}{2}\left(6\sqrt{3}-6\sqrt{3}+\sqrt{3}t\right)$$

Multiply \frac{1}{2} and -1 to get -\frac{1}{2}.

$$\left(3-\frac{1}{2}t\right)\left(6\sqrt{3}-\sqrt{3}t\right)+\frac{6}{2}\left(6\sqrt{3}-6\sqrt{3}+\sqrt{3}t\right)$$

Use the distributive property to multiply 3-\frac{1}{2}t by 6\sqrt{3}-\sqrt{3}t.

$$3\left(6\sqrt{3}-\sqrt{3}t\right)-\frac{1}{2}t\left(6\sqrt{3}-\sqrt{3}t\right)+\frac{6}{2}\left(6\sqrt{3}-6\sqrt{3}+\sqrt{3}t\right)$$

Divide 6 by 2 to get 3.

$$3\left(6\sqrt{3}-\sqrt{3}t\right)-\frac{1}{2}t\left(6\sqrt{3}-\sqrt{3}t\right)+3\left(6\sqrt{3}-6\sqrt{3}+\sqrt{3}t\right)$$

Combine 6\sqrt{3} and -6\sqrt{3} to get 0.

$$3\left(6\sqrt{3}-\sqrt{3}t\right)-\frac{1}{2}t\left(6\sqrt{3}-\sqrt{3}t\right)+3\sqrt{3}t$$

Use the distributive property to multiply 3 by 6\sqrt{3}-\sqrt{3}t.

$$18\sqrt{3}-3\sqrt{3}t-\frac{1}{2}t\left(6\sqrt{3}-\sqrt{3}t\right)+3\sqrt{3}t$$

Use the distributive property to multiply -\frac{1}{2}t by 6\sqrt{3}-\sqrt{3}t.

$$18\sqrt{3}-3\sqrt{3}t-\frac{1}{2}t\times 6\sqrt{3}-\frac{1}{2}t\left(-1\right)\sqrt{3}t+3\sqrt{3}t$$

Multiply t and t to get t^{2}.

$$18\sqrt{3}-3\sqrt{3}t-\frac{1}{2}t\times 6\sqrt{3}-\frac{1}{2}t^{2}\left(-1\right)\sqrt{3}+3\sqrt{3}t$$

Express -\frac{1}{2}\times 6 as a single fraction.

$$18\sqrt{3}-3\sqrt{3}t+\frac{-6}{2}t\sqrt{3}-\frac{1}{2}t^{2}\left(-1\right)\sqrt{3}+3\sqrt{3}t$$

Divide -6 by 2 to get -3.

$$18\sqrt{3}-3\sqrt{3}t-3t\sqrt{3}-\frac{1}{2}t^{2}\left(-1\right)\sqrt{3}+3\sqrt{3}t$$

Multiply -\frac{1}{2} and -1 to get \frac{1}{2}.

$$18\sqrt{3}-3\sqrt{3}t-3t\sqrt{3}+\frac{1}{2}t^{2}\sqrt{3}+3\sqrt{3}t$$

Combine -3\sqrt{3}t and -3t\sqrt{3} to get -6\sqrt{3}t.

$$18\sqrt{3}-6\sqrt{3}t+\frac{1}{2}t^{2}\sqrt{3}+3\sqrt{3}t$$

Combine -6\sqrt{3}t and 3\sqrt{3}t to get -3\sqrt{3}t.

$$18\sqrt{3}-3\sqrt{3}t+\frac{1}{2}t^{2}\sqrt{3}$$

Factor

$$\frac{\sqrt{3}\left(t^{2}-6t+36\right)}{2}$$

Solution Steps

Factor out \frac{1}{2}.

$$\frac{\left(6-t\right)\left(6\sqrt{3}-\sqrt{3}t\right)+6\left(6\sqrt{3}-6\sqrt{3}+\sqrt{3}t\right)}{2}$$