Solve (3sqrt{2}-1)(1+3sqrt{2/4})2/2-(2sqrt{2}-1)^2

$$( 3 \sqrt { 2 } - 1 ) ( 1 + 3 \sqrt { \frac { 2 } { 4 } } ) \frac { 2 } { 2 } - ( 2 \sqrt { 2 } - 1 ) ^ { 2 }$$

Evaluate

$$\frac{11\sqrt{2}}{2}-1\approx 6.778174593$$

Short Solution Steps

Divide 2 by 2 to get 1.

$$\left(3\sqrt{2}-1\right)\left(1+3\sqrt{\frac{2}{4}}\right)\times 1-\left(2\sqrt{2}-1\right)^{2}$$

Reduce the fraction \frac{2}{4} to lowest terms by extracting and canceling out 2.

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

Rewrite the square root of the division \sqrt{\frac{1}{2}} as the division of square roots \frac{\sqrt{1}}{\sqrt{2}}.

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

Calculate the square root of 1 and get 1.

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

Rationalize the denominator of \frac{1}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

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

The square of \sqrt{2} is 2.

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

Express 3\times \frac{\sqrt{2}}{2} as a single fraction.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{2}{2}.

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

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

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

Express \left(3\sqrt{2}-1\right)\times \frac{2+3\sqrt{2}}{2} as a single fraction.

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

Express \frac{\left(3\sqrt{2}-1\right)\left(2+3\sqrt{2}\right)}{2}\times 1 as a single fraction.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2\sqrt{2}-1\right)^{2}.

$$\frac{\left(3\sqrt{2}-1\right)\left(2+3\sqrt{2}\right)}{2}-\left(4\left(\sqrt{2}\right)^{2}-4\sqrt{2}+1\right)$$

The square of \sqrt{2} is 2.

$$\frac{\left(3\sqrt{2}-1\right)\left(2+3\sqrt{2}\right)}{2}-\left(4\times 2-4\sqrt{2}+1\right)$$

Multiply 4 and 2 to get 8.

$$\frac{\left(3\sqrt{2}-1\right)\left(2+3\sqrt{2}\right)}{2}-\left(8-4\sqrt{2}+1\right)$$

Add 8 and 1 to get 9.

$$\frac{\left(3\sqrt{2}-1\right)\left(2+3\sqrt{2}\right)}{2}-\left(9-4\sqrt{2}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 9-4\sqrt{2} times \frac{2}{2}.

$$\frac{\left(3\sqrt{2}-1\right)\left(2+3\sqrt{2}\right)}{2}-\frac{2\left(9-4\sqrt{2}\right)}{2}$$

Since \frac{\left(3\sqrt{2}-1\right)\left(2+3\sqrt{2}\right)}{2} and \frac{2\left(9-4\sqrt{2}\right)}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(3\sqrt{2}-1\right)\left(2+3\sqrt{2}\right)-2\left(9-4\sqrt{2}\right)}{2}$$

Do the multiplications in \left(3\sqrt{2}-1\right)\left(2+3\sqrt{2}\right)-2\left(9-4\sqrt{2}\right).

$$\frac{6\sqrt{2}+18-2-3\sqrt{2}-18+8\sqrt{2}}{2}$$

Do the calculations in 6\sqrt{2}+18-2-3\sqrt{2}-18+8\sqrt{2}.

$$\frac{11\sqrt{2}-2}{2}$$