Solve (x/4-1/2)(3-x)-(3-x)(3x/4+7/2)

$$( \frac { x } { 4 } - \frac { 1 } { 2 } ) ( 3 - x ) - ( 3 - x ) ( \frac { 3 x } { 4 } + \frac { 7 } { 2 } )$$

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4 and 2 is 4. Multiply \frac{1}{2} times \frac{2}{2}.

$$\left(\frac{x}{4}-\frac{2}{4}\right)\left(3-x\right)-\left(3-x\right)\left(\frac{3x}{4}+\frac{7}{2}\right)$$

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

$$\frac{x-2}{4}\left(3-x\right)-\left(3-x\right)\left(\frac{3x}{4}+\frac{7}{2}\right)$$

Express \frac{x-2}{4}\left(3-x\right) as a single fraction.

$$\frac{\left(x-2\right)\left(3-x\right)}{4}-\left(3-x\right)\left(\frac{3x}{4}+\frac{7}{2}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4 and 2 is 4. Multiply \frac{7}{2} times \frac{2}{2}.

$$\frac{\left(x-2\right)\left(3-x\right)}{4}-\left(3-x\right)\left(\frac{3x}{4}+\frac{7\times 2}{4}\right)$$

Since \frac{3x}{4} and \frac{7\times 2}{4} have the same denominator, add them by adding their numerators.

$$\frac{\left(x-2\right)\left(3-x\right)}{4}-\left(3-x\right)\times \frac{3x+7\times 2}{4}$$

Do the multiplications in 3x+7\times 2.

$$\frac{\left(x-2\right)\left(3-x\right)}{4}-\left(3-x\right)\times \frac{3x+14}{4}$$

Express \left(3-x\right)\times \frac{3x+14}{4} as a single fraction.

$$\frac{\left(x-2\right)\left(3-x\right)}{4}-\frac{\left(3-x\right)\left(3x+14\right)}{4}$$

Since \frac{\left(x-2\right)\left(3-x\right)}{4} and \frac{\left(3-x\right)\left(3x+14\right)}{4} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x-2\right)\left(3-x\right)-\left(3-x\right)\left(3x+14\right)}{4}$$

Do the multiplications in \left(x-2\right)\left(3-x\right)-\left(3-x\right)\left(3x+14\right).

$$\frac{3x-x^{2}-6+2x-9x-42+3x^{2}+14x}{4}$$

Combine like terms in 3x-x^{2}-6+2x-9x-42+3x^{2}+14x.

$$\frac{10x+2x^{2}-48}{4}$$

Expand

$$\frac{x^{2}}{2}+\frac{5x}{2}-12$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4 and 2 is 4. Multiply \frac{1}{2} times \frac{2}{2}.

$$\left(\frac{x}{4}-\frac{2}{4}\right)\left(3-x\right)-\left(3-x\right)\left(\frac{3x}{4}+\frac{7}{2}\right)$$

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

$$\frac{x-2}{4}\left(3-x\right)-\left(3-x\right)\left(\frac{3x}{4}+\frac{7}{2}\right)$$

Express \frac{x-2}{4}\left(3-x\right) as a single fraction.

$$\frac{\left(x-2\right)\left(3-x\right)}{4}-\left(3-x\right)\left(\frac{3x}{4}+\frac{7}{2}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4 and 2 is 4. Multiply \frac{7}{2} times \frac{2}{2}.

$$\frac{\left(x-2\right)\left(3-x\right)}{4}-\left(3-x\right)\left(\frac{3x}{4}+\frac{7\times 2}{4}\right)$$

Since \frac{3x}{4} and \frac{7\times 2}{4} have the same denominator, add them by adding their numerators.

$$\frac{\left(x-2\right)\left(3-x\right)}{4}-\left(3-x\right)\times \frac{3x+7\times 2}{4}$$

Do the multiplications in 3x+7\times 2.

$$\frac{\left(x-2\right)\left(3-x\right)}{4}-\left(3-x\right)\times \frac{3x+14}{4}$$

Express \left(3-x\right)\times \frac{3x+14}{4} as a single fraction.

$$\frac{\left(x-2\right)\left(3-x\right)}{4}-\frac{\left(3-x\right)\left(3x+14\right)}{4}$$

Since \frac{\left(x-2\right)\left(3-x\right)}{4} and \frac{\left(3-x\right)\left(3x+14\right)}{4} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x-2\right)\left(3-x\right)-\left(3-x\right)\left(3x+14\right)}{4}$$

Do the multiplications in \left(x-2\right)\left(3-x\right)-\left(3-x\right)\left(3x+14\right).

$$\frac{3x-x^{2}-6+2x-9x-42+3x^{2}+14x}{4}$$

Combine like terms in 3x-x^{2}-6+2x-9x-42+3x^{2}+14x.

$$\frac{10x+2x^{2}-48}{4}$$