Solve (10-sqrt{2}c+12)*(frac{sqrt{2}}{2}c+10)*1/2

$$( 10 - \sqrt { 2 } c + 12 ) \times ( \frac { \sqrt { 2 } } { 2 } c + 10 ) \times \frac { 1 } { 2 }$$

Evaluate

$$\frac{-c^{2}+\sqrt{2}c+220}{2}$$

Short Solution Steps

Express \frac{\sqrt{2}}{2}c as a single fraction.

$$\left(10-\sqrt{2}c+12\right)\left(\frac{\sqrt{2}c}{2}+10\right)\times \frac{1}{2}$$

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

$$\left(10-\sqrt{2}c+12\right)\left(\frac{\sqrt{2}c}{2}+\frac{10\times 2}{2}\right)\times \frac{1}{2}$$

Since \frac{\sqrt{2}c}{2} and \frac{10\times 2}{2} have the same denominator, add them by adding their numerators.

$$\left(10-\sqrt{2}c+12\right)\times \frac{\sqrt{2}c+10\times 2}{2}\times \frac{1}{2}$$

Do the multiplications in \sqrt{2}c+10\times 2.

$$\left(10-\sqrt{2}c+12\right)\times \frac{\sqrt{2}c+20}{2}\times \frac{1}{2}$$

Express \left(10-\sqrt{2}c+12\right)\times \frac{\sqrt{2}c+20}{2} as a single fraction.

$$\frac{\left(10-\sqrt{2}c+12\right)\left(\sqrt{2}c+20\right)}{2}\times \frac{1}{2}$$

Multiply \frac{\left(10-\sqrt{2}c+12\right)\left(\sqrt{2}c+20\right)}{2} times \frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(10-\sqrt{2}c+12\right)\left(\sqrt{2}c+20\right)}{2\times 2}$$

Multiply 2 and 2 to get 4.

$$\frac{\left(10-\sqrt{2}c+12\right)\left(\sqrt{2}c+20\right)}{4}$$

Add 10 and 12 to get 22.

$$\frac{\left(22-\sqrt{2}c\right)\left(\sqrt{2}c+20\right)}{4}$$

Apply the distributive property by multiplying each term of 22-\sqrt{2}c by each term of \sqrt{2}c+20.

$$\frac{22\sqrt{2}c+440-\left(\sqrt{2}\right)^{2}c^{2}-20c\sqrt{2}}{4}$$

The square of \sqrt{2} is 2.

$$\frac{22\sqrt{2}c+440-2c^{2}-20c\sqrt{2}}{4}$$

Combine 22\sqrt{2}c and -20c\sqrt{2} to get 2\sqrt{2}c.

$$\frac{2\sqrt{2}c+440-2c^{2}}{4}$$

Factor

$$\frac{-c^{2}+\sqrt{2}c+220}{2}$$