Solve (5sqrt{2}-4)^2-(3-sqrt{2})^2

$$( 5 \sqrt { 2 } - 4 ) ^ { 2 } - ( 3 - \sqrt { 2 } ) ^ { 2 }$$

Evaluate

$$55-34\sqrt{2}\approx 6.916738879$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5\sqrt{2}-4\right)^{2}.

$$25\left(\sqrt{2}\right)^{2}-40\sqrt{2}+16-\left(3-\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$25\times 2-40\sqrt{2}+16-\left(3-\sqrt{2}\right)^{2}$$

Multiply 25 and 2 to get 50.

$$50-40\sqrt{2}+16-\left(3-\sqrt{2}\right)^{2}$$

Add 50 and 16 to get 66.

$$66-40\sqrt{2}-\left(3-\sqrt{2}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3-\sqrt{2}\right)^{2}.

$$66-40\sqrt{2}-\left(9-6\sqrt{2}+\left(\sqrt{2}\right)^{2}\right)$$

The square of \sqrt{2} is 2.

$$66-40\sqrt{2}-\left(9-6\sqrt{2}+2\right)$$

Add 9 and 2 to get 11.

$$66-40\sqrt{2}-\left(11-6\sqrt{2}\right)$$

To find the opposite of 11-6\sqrt{2}, find the opposite of each term.

$$66-40\sqrt{2}-11+6\sqrt{2}$$

Subtract 11 from 66 to get 55.

$$55-40\sqrt{2}+6\sqrt{2}$$

Combine -40\sqrt{2} and 6\sqrt{2} to get -34\sqrt{2}.

$$55-34\sqrt{2}$$

Expand

$$55-34\sqrt{2}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5\sqrt{2}-4\right)^{2}.

$$25\left(\sqrt{2}\right)^{2}-40\sqrt{2}+16-\left(3-\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$25\times 2-40\sqrt{2}+16-\left(3-\sqrt{2}\right)^{2}$$

Multiply 25 and 2 to get 50.

$$50-40\sqrt{2}+16-\left(3-\sqrt{2}\right)^{2}$$

Add 50 and 16 to get 66.

$$66-40\sqrt{2}-\left(3-\sqrt{2}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3-\sqrt{2}\right)^{2}.

$$66-40\sqrt{2}-\left(9-6\sqrt{2}+\left(\sqrt{2}\right)^{2}\right)$$

The square of \sqrt{2} is 2.

$$66-40\sqrt{2}-\left(9-6\sqrt{2}+2\right)$$

Add 9 and 2 to get 11.

$$66-40\sqrt{2}-\left(11-6\sqrt{2}\right)$$

To find the opposite of 11-6\sqrt{2}, find the opposite of each term.

$$66-40\sqrt{2}-11+6\sqrt{2}$$

Subtract 11 from 66 to get 55.

$$55-40\sqrt{2}+6\sqrt{2}$$

Combine -40\sqrt{2} and 6\sqrt{2} to get -34\sqrt{2}.

$$55-34\sqrt{2}$$