Solve (x/2-2/5)(2/5-x/2)-x^2+2x

$$( \frac { x } { 2 } - \frac { 2 } { 5 } ) ( \frac { 2 } { 5 } - \frac { x } { 2 } ) - x ^ { 2 } + 2 x$$

Evaluate

$$-\frac{5x^{2}}{4}+\frac{12x}{5}-\frac{4}{25}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 5 is 10. Multiply \frac{x}{2} times \frac{5}{5}. Multiply \frac{2}{5} times \frac{2}{2}.

$$\left(\frac{5x}{10}-\frac{2\times 2}{10}\right)\left(\frac{2}{5}-\frac{x}{2}\right)-x^{2}+2x$$

Since \frac{5x}{10} and \frac{2\times 2}{10} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5x-2\times 2}{10}\left(\frac{2}{5}-\frac{x}{2}\right)-x^{2}+2x$$

Do the multiplications in 5x-2\times 2.

$$\frac{5x-4}{10}\left(\frac{2}{5}-\frac{x}{2}\right)-x^{2}+2x$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5 and 2 is 10. Multiply \frac{2}{5} times \frac{2}{2}. Multiply \frac{x}{2} times \frac{5}{5}.

$$\frac{5x-4}{10}\left(\frac{2\times 2}{10}-\frac{5x}{10}\right)-x^{2}+2x$$

Since \frac{2\times 2}{10} and \frac{5x}{10} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5x-4}{10}\times \frac{2\times 2-5x}{10}-x^{2}+2x$$

Do the multiplications in 2\times 2-5x.

$$\frac{5x-4}{10}\times \frac{4-5x}{10}-x^{2}+2x$$

Multiply \frac{5x-4}{10} times \frac{4-5x}{10} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(5x-4\right)\left(4-5x\right)}{10\times 10}-x^{2}+2x$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -x^{2}+2x times \frac{10\times 10}{10\times 10}.

$$\frac{\left(5x-4\right)\left(4-5x\right)}{10\times 10}+\frac{\left(-x^{2}+2x\right)\times 10\times 10}{10\times 10}$$

Since \frac{\left(5x-4\right)\left(4-5x\right)}{10\times 10} and \frac{\left(-x^{2}+2x\right)\times 10\times 10}{10\times 10} have the same denominator, add them by adding their numerators.

$$\frac{\left(5x-4\right)\left(4-5x\right)+\left(-x^{2}+2x\right)\times 10\times 10}{10\times 10}$$

Do the multiplications in \left(5x-4\right)\left(4-5x\right)+\left(-x^{2}+2x\right)\times 10\times 10.

$$\frac{20x-25x^{2}-16+20x-100x^{2}+200x}{10\times 10}$$

Combine like terms in 20x-25x^{2}-16+20x-100x^{2}+200x.

$$\frac{240x-125x^{2}-16}{10\times 10}$$

Factor the expressions that are not already factored in \frac{240x-125x^{2}-16}{10\times 10}.

$$\frac{-125\left(x-\left(-\frac{4}{25}\sqrt{31}+\frac{24}{25}\right)\right)\left(x-\left(\frac{4}{25}\sqrt{31}+\frac{24}{25}\right)\right)}{10\times 10}$$

Cancel out 5\times 5 in both numerator and denominator.

$$\frac{-5\left(x-\left(-\frac{4}{25}\sqrt{31}+\frac{24}{25}\right)\right)\left(x-\left(\frac{4}{25}\sqrt{31}+\frac{24}{25}\right)\right)}{2\times 2}$$

Expand 2\times 2.

$$\frac{-5\left(x-\left(-\frac{4}{25}\sqrt{31}+\frac{24}{25}\right)\right)\left(x-\left(\frac{4}{25}\sqrt{31}+\frac{24}{25}\right)\right)}{4}$$

To find the opposite of -\frac{4}{25}\sqrt{31}+\frac{24}{25}, find the opposite of each term.

$$\frac{-5\left(x+\frac{4}{25}\sqrt{31}-\frac{24}{25}\right)\left(x-\left(\frac{4}{25}\sqrt{31}+\frac{24}{25}\right)\right)}{4}$$

To find the opposite of \frac{4}{25}\sqrt{31}+\frac{24}{25}, find the opposite of each term.

$$\frac{-5\left(x+\frac{4}{25}\sqrt{31}-\frac{24}{25}\right)\left(x-\frac{4}{25}\sqrt{31}-\frac{24}{25}\right)}{4}$$

Use the distributive property to multiply -5 by x+\frac{4}{25}\sqrt{31}-\frac{24}{25}.

$$\frac{\left(-5x-\frac{4}{5}\sqrt{31}+\frac{24}{5}\right)\left(x-\frac{4}{25}\sqrt{31}-\frac{24}{25}\right)}{4}$$

Use the distributive property to multiply -5x-\frac{4}{5}\sqrt{31}+\frac{24}{5} by x-\frac{4}{25}\sqrt{31}-\frac{24}{25} and combine like terms.

$$\frac{-5x^{2}+\frac{48}{5}x+\frac{16}{125}\left(\sqrt{31}\right)^{2}-\frac{576}{125}}{4}$$

The square of \sqrt{31} is 31.

$$\frac{-5x^{2}+\frac{48}{5}x+\frac{16}{125}\times 31-\frac{576}{125}}{4}$$

Multiply \frac{16}{125} and 31 to get \frac{496}{125}.

$$\frac{-5x^{2}+\frac{48}{5}x+\frac{496}{125}-\frac{576}{125}}{4}$$

Subtract \frac{576}{125} from \frac{496}{125} to get -\frac{16}{25}.

$$\frac{-5x^{2}+\frac{48}{5}x-\frac{16}{25}}{4}$$

Expand

$$-\frac{5x^{2}}{4}+\frac{12x}{5}-\frac{4}{25}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 5 is 10. Multiply \frac{x}{2} times \frac{5}{5}. Multiply \frac{2}{5} times \frac{2}{2}.

$$\left(\frac{5x}{10}-\frac{2\times 2}{10}\right)\left(\frac{2}{5}-\frac{x}{2}\right)-x^{2}+2x$$

Since \frac{5x}{10} and \frac{2\times 2}{10} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5x-2\times 2}{10}\left(\frac{2}{5}-\frac{x}{2}\right)-x^{2}+2x$$

Do the multiplications in 5x-2\times 2.

$$\frac{5x-4}{10}\left(\frac{2}{5}-\frac{x}{2}\right)-x^{2}+2x$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5 and 2 is 10. Multiply \frac{2}{5} times \frac{2}{2}. Multiply \frac{x}{2} times \frac{5}{5}.

$$\frac{5x-4}{10}\left(\frac{2\times 2}{10}-\frac{5x}{10}\right)-x^{2}+2x$$

Since \frac{2\times 2}{10} and \frac{5x}{10} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5x-4}{10}\times \frac{2\times 2-5x}{10}-x^{2}+2x$$

Do the multiplications in 2\times 2-5x.

$$\frac{5x-4}{10}\times \frac{4-5x}{10}-x^{2}+2x$$

Multiply \frac{5x-4}{10} times \frac{4-5x}{10} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(5x-4\right)\left(4-5x\right)}{10\times 10}-x^{2}+2x$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -x^{2}+2x times \frac{10\times 10}{10\times 10}.

$$\frac{\left(5x-4\right)\left(4-5x\right)}{10\times 10}+\frac{\left(-x^{2}+2x\right)\times 10\times 10}{10\times 10}$$

Since \frac{\left(5x-4\right)\left(4-5x\right)}{10\times 10} and \frac{\left(-x^{2}+2x\right)\times 10\times 10}{10\times 10} have the same denominator, add them by adding their numerators.

$$\frac{\left(5x-4\right)\left(4-5x\right)+\left(-x^{2}+2x\right)\times 10\times 10}{10\times 10}$$

Do the multiplications in \left(5x-4\right)\left(4-5x\right)+\left(-x^{2}+2x\right)\times 10\times 10.

$$\frac{20x-25x^{2}-16+20x-100x^{2}+200x}{10\times 10}$$

Combine like terms in 20x-25x^{2}-16+20x-100x^{2}+200x.

$$\frac{240x-125x^{2}-16}{10\times 10}$$

Factor the expressions that are not already factored in \frac{240x-125x^{2}-16}{10\times 10}.

$$\frac{-125\left(x-\left(-\frac{4}{25}\sqrt{31}+\frac{24}{25}\right)\right)\left(x-\left(\frac{4}{25}\sqrt{31}+\frac{24}{25}\right)\right)}{10\times 10}$$

Cancel out 5\times 5 in both numerator and denominator.

$$\frac{-5\left(x-\left(-\frac{4}{25}\sqrt{31}+\frac{24}{25}\right)\right)\left(x-\left(\frac{4}{25}\sqrt{31}+\frac{24}{25}\right)\right)}{2\times 2}$$

Expand 2\times 2.

$$\frac{-5\left(x-\left(-\frac{4}{25}\sqrt{31}+\frac{24}{25}\right)\right)\left(x-\left(\frac{4}{25}\sqrt{31}+\frac{24}{25}\right)\right)}{4}$$

To find the opposite of -\frac{4}{25}\sqrt{31}+\frac{24}{25}, find the opposite of each term.

$$\frac{-5\left(x+\frac{4}{25}\sqrt{31}-\frac{24}{25}\right)\left(x-\left(\frac{4}{25}\sqrt{31}+\frac{24}{25}\right)\right)}{4}$$

To find the opposite of \frac{4}{25}\sqrt{31}+\frac{24}{25}, find the opposite of each term.

$$\frac{-5\left(x+\frac{4}{25}\sqrt{31}-\frac{24}{25}\right)\left(x-\frac{4}{25}\sqrt{31}-\frac{24}{25}\right)}{4}$$

Use the distributive property to multiply -5 by x+\frac{4}{25}\sqrt{31}-\frac{24}{25}.

$$\frac{\left(-5x-\frac{4}{5}\sqrt{31}+\frac{24}{5}\right)\left(x-\frac{4}{25}\sqrt{31}-\frac{24}{25}\right)}{4}$$

Use the distributive property to multiply -5x-\frac{4}{5}\sqrt{31}+\frac{24}{5} by x-\frac{4}{25}\sqrt{31}-\frac{24}{25} and combine like terms.

$$\frac{-5x^{2}+\frac{48}{5}x+\frac{16}{125}\left(\sqrt{31}\right)^{2}-\frac{576}{125}}{4}$$

The square of \sqrt{31} is 31.

$$\frac{-5x^{2}+\frac{48}{5}x+\frac{16}{125}\times 31-\frac{576}{125}}{4}$$

Multiply \frac{16}{125} and 31 to get \frac{496}{125}.

$$\frac{-5x^{2}+\frac{48}{5}x+\frac{496}{125}-\frac{576}{125}}{4}$$

Subtract \frac{576}{125} from \frac{496}{125} to get -\frac{16}{25}.

$$\frac{-5x^{2}+\frac{48}{5}x-\frac{16}{25}}{4}$$