Solve 4sqrt{3}-(2-x/4)(4sqrt{3}-frac{sqrt{3}x}{2})-(x/2-2)(frac{sqrt{3}x}{2}-2sqrt{3}

$$4 \sqrt{ 3 } -(2- \frac{ x }{ 4 } )(4 \sqrt{ 3 } - \frac{ \sqrt{ 3 } x }{ 2 } )-( \frac{ x }{ 2 } -2)( \frac{ \sqrt{ 3 } x }{ 2 } -2 \sqrt{ 3 }$$

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{4}{4}.

$$4\sqrt{3}-\left(\frac{2\times 4}{4}-\frac{x}{4}\right)\left(4\sqrt{3}-\frac{\sqrt{3}x}{2}\right)-\left(\frac{x}{2}-2\right)\left(\frac{\sqrt{3}x}{2}-2\sqrt{3}\right)$$

Since \frac{2\times 4}{4} and \frac{x}{4} have the same denominator, subtract them by subtracting their numerators.

$$4\sqrt{3}-\frac{2\times 4-x}{4}\left(4\sqrt{3}-\frac{\sqrt{3}x}{2}\right)-\left(\frac{x}{2}-2\right)\left(\frac{\sqrt{3}x}{2}-2\sqrt{3}\right)$$

Do the multiplications in 2\times 4-x.

$$4\sqrt{3}-\frac{8-x}{4}\left(4\sqrt{3}-\frac{\sqrt{3}x}{2}\right)-\left(\frac{x}{2}-2\right)\left(\frac{\sqrt{3}x}{2}-2\sqrt{3}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 4\sqrt{3} times \frac{2}{2}.

$$4\sqrt{3}-\frac{8-x}{4}\left(\frac{2\times 4\sqrt{3}}{2}-\frac{\sqrt{3}x}{2}\right)-\left(\frac{x}{2}-2\right)\left(\frac{\sqrt{3}x}{2}-2\sqrt{3}\right)$$

Since \frac{2\times 4\sqrt{3}}{2} and \frac{\sqrt{3}x}{2} have the same denominator, subtract them by subtracting their numerators.

$$4\sqrt{3}-\frac{8-x}{4}\times \frac{2\times 4\sqrt{3}-\sqrt{3}x}{2}-\left(\frac{x}{2}-2\right)\left(\frac{\sqrt{3}x}{2}-2\sqrt{3}\right)$$

Do the multiplications in 2\times 4\sqrt{3}-\sqrt{3}x.

$$4\sqrt{3}-\frac{8-x}{4}\times \frac{8\sqrt{3}-\sqrt{3}x}{2}-\left(\frac{x}{2}-2\right)\left(\frac{\sqrt{3}x}{2}-2\sqrt{3}\right)$$

Multiply \frac{8-x}{4} times \frac{8\sqrt{3}-\sqrt{3}x}{2} by multiplying numerator times numerator and denominator times denominator.

$$4\sqrt{3}-\frac{\left(8-x\right)\left(8\sqrt{3}-\sqrt{3}x\right)}{4\times 2}-\left(\frac{x}{2}-2\right)\left(\frac{\sqrt{3}x}{2}-2\sqrt{3}\right)$$

Multiply 4 and 2 to get 8.

$$4\sqrt{3}-\frac{\left(8-x\right)\left(8\sqrt{3}-\sqrt{3}x\right)}{8}-\left(\frac{x}{2}-2\right)\left(\frac{\sqrt{3}x}{2}-2\sqrt{3}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 4\sqrt{3} times \frac{8}{8}.

$$\frac{8\times 4\sqrt{3}}{8}-\frac{\left(8-x\right)\left(8\sqrt{3}-\sqrt{3}x\right)}{8}-\left(\frac{x}{2}-2\right)\left(\frac{\sqrt{3}x}{2}-2\sqrt{3}\right)$$

Since \frac{8\times 4\sqrt{3}}{8} and \frac{\left(8-x\right)\left(8\sqrt{3}-\sqrt{3}x\right)}{8} have the same denominator, subtract them by subtracting their numerators.

$$\frac{8\times 4\sqrt{3}-\left(8-x\right)\left(8\sqrt{3}-\sqrt{3}x\right)}{8}-\left(\frac{x}{2}-2\right)\left(\frac{\sqrt{3}x}{2}-2\sqrt{3}\right)$$

Do the multiplications in 8\times 4\sqrt{3}-\left(8-x\right)\left(8\sqrt{3}-\sqrt{3}x\right).

$$\frac{32\sqrt{3}-64\sqrt{3}+8\sqrt{3}x+8\sqrt{3}x-x^{2}\sqrt{3}}{8}-\left(\frac{x}{2}-2\right)\left(\frac{\sqrt{3}x}{2}-2\sqrt{3}\right)$$

Combine like terms in 32\sqrt{3}-64\sqrt{3}+8\sqrt{3}x+8\sqrt{3}x-x^{2}\sqrt{3}.

$$\frac{-x^{2}\sqrt{3}-32\sqrt{3}+16\sqrt{3}x}{8}-\left(\frac{x}{2}-2\right)\left(\frac{\sqrt{3}x}{2}-2\sqrt{3}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{2}{2}.

$$\frac{-x^{2}\sqrt{3}-32\sqrt{3}+16\sqrt{3}x}{8}-\left(\frac{x}{2}-\frac{2\times 2}{2}\right)\left(\frac{\sqrt{3}x}{2}-2\sqrt{3}\right)$$

Since \frac{x}{2} and \frac{2\times 2}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-x^{2}\sqrt{3}-32\sqrt{3}+16\sqrt{3}x}{8}-\frac{x-2\times 2}{2}\left(\frac{\sqrt{3}x}{2}-2\sqrt{3}\right)$$

Do the multiplications in x-2\times 2.

$$\frac{-x^{2}\sqrt{3}-32\sqrt{3}+16\sqrt{3}x}{8}-\frac{x-4}{2}\left(\frac{\sqrt{3}x}{2}-2\sqrt{3}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -2\sqrt{3} times \frac{2}{2}.

$$\frac{-x^{2}\sqrt{3}-32\sqrt{3}+16\sqrt{3}x}{8}-\frac{x-4}{2}\left(\frac{\sqrt{3}x}{2}+\frac{2\left(-2\right)\sqrt{3}}{2}\right)$$

Since \frac{\sqrt{3}x}{2} and \frac{2\left(-2\right)\sqrt{3}}{2} have the same denominator, add them by adding their numerators.

$$\frac{-x^{2}\sqrt{3}-32\sqrt{3}+16\sqrt{3}x}{8}-\frac{x-4}{2}\times \frac{\sqrt{3}x+2\left(-2\right)\sqrt{3}}{2}$$

Do the multiplications in \sqrt{3}x+2\left(-2\right)\sqrt{3}.

$$\frac{-x^{2}\sqrt{3}-32\sqrt{3}+16\sqrt{3}x}{8}-\frac{x-4}{2}\times \frac{\sqrt{3}x-4\sqrt{3}}{2}$$

Multiply \frac{x-4}{2} times \frac{\sqrt{3}x-4\sqrt{3}}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-x^{2}\sqrt{3}-32\sqrt{3}+16\sqrt{3}x}{8}-\frac{\left(x-4\right)\left(\sqrt{3}x-4\sqrt{3}\right)}{2\times 2}$$

Multiply 2 and 2 to get 4.

$$\frac{-x^{2}\sqrt{3}-32\sqrt{3}+16\sqrt{3}x}{8}-\frac{\left(x-4\right)\left(\sqrt{3}x-4\sqrt{3}\right)}{4}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 8 and 4 is 8. Multiply \frac{\left(x-4\right)\left(\sqrt{3}x-4\sqrt{3}\right)}{4} times \frac{2}{2}.

$$\frac{-x^{2}\sqrt{3}-32\sqrt{3}+16\sqrt{3}x}{8}-\frac{2\left(x-4\right)\left(\sqrt{3}x-4\sqrt{3}\right)}{8}$$

Since \frac{-x^{2}\sqrt{3}-32\sqrt{3}+16\sqrt{3}x}{8} and \frac{2\left(x-4\right)\left(\sqrt{3}x-4\sqrt{3}\right)}{8} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-x^{2}\sqrt{3}-32\sqrt{3}+16\sqrt{3}x-2\left(x-4\right)\left(\sqrt{3}x-4\sqrt{3}\right)}{8}$$

Do the multiplications in -x^{2}\sqrt{3}-32\sqrt{3}+16\sqrt{3}x-2\left(x-4\right)\left(\sqrt{3}x-4\sqrt{3}\right).

$$\frac{-x^{2}\sqrt{3}-32\sqrt{3}+16\sqrt{3}x-2x^{2}\sqrt{3}+8x\sqrt{3}+8x\sqrt{3}-32\sqrt{3}}{8}$$

Combine like terms in -x^{2}\sqrt{3}-32\sqrt{3}+16\sqrt{3}x-2x^{2}\sqrt{3}+8x\sqrt{3}+8x\sqrt{3}-32\sqrt{3}.

$$\frac{-64\sqrt{3}-3x^{2}\sqrt{3}+32\sqrt{3}x}{8}$$

Factor

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

Solution Steps

Factor out \frac{1}{8}.

$$\frac{32\sqrt{3}-\left(8-x\right)\left(8\sqrt{3}-\sqrt{3}x\right)-2\left(x-4\right)\left(\sqrt{3}x-4\sqrt{3}\right)}{8}$$