Solve (3sqrt{2}+2)^2-(2-2sqrt{2})^2

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

Evaluate

$$20\sqrt{2}+10\approx 38.284271247$$

Solution Steps

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

$$9\left(\sqrt{2}\right)^{2}+12\sqrt{2}+4-\left(2-2\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$9\times 2+12\sqrt{2}+4-\left(2-2\sqrt{2}\right)^{2}$$

Multiply 9 and 2 to get 18.

$$18+12\sqrt{2}+4-\left(2-2\sqrt{2}\right)^{2}$$

Add 18 and 4 to get 22.

$$22+12\sqrt{2}-\left(2-2\sqrt{2}\right)^{2}$$

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

$$22+12\sqrt{2}-\left(4-8\sqrt{2}+4\left(\sqrt{2}\right)^{2}\right)$$

The square of \sqrt{2} is 2.

$$22+12\sqrt{2}-\left(4-8\sqrt{2}+4\times 2\right)$$

Multiply 4 and 2 to get 8.

$$22+12\sqrt{2}-\left(4-8\sqrt{2}+8\right)$$

Add 4 and 8 to get 12.

$$22+12\sqrt{2}-\left(12-8\sqrt{2}\right)$$

To find the opposite of 12-8\sqrt{2}, find the opposite of each term.

$$22+12\sqrt{2}-12+8\sqrt{2}$$

Subtract 12 from 22 to get 10.

$$10+12\sqrt{2}+8\sqrt{2}$$

Combine 12\sqrt{2} and 8\sqrt{2} to get 20\sqrt{2}.

$$10+20\sqrt{2}$$

Expand

$$20 \sqrt{2} + 10 = 38.284271247$$

Solution Steps

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

$$9\left(\sqrt{2}\right)^{2}+12\sqrt{2}+4-\left(2-2\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$9\times 2+12\sqrt{2}+4-\left(2-2\sqrt{2}\right)^{2}$$

Multiply 9 and 2 to get 18.

$$18+12\sqrt{2}+4-\left(2-2\sqrt{2}\right)^{2}$$

Add 18 and 4 to get 22.

$$22+12\sqrt{2}-\left(2-2\sqrt{2}\right)^{2}$$

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

$$22+12\sqrt{2}-\left(4-8\sqrt{2}+4\left(\sqrt{2}\right)^{2}\right)$$

The square of \sqrt{2} is 2.

$$22+12\sqrt{2}-\left(4-8\sqrt{2}+4\times 2\right)$$

Multiply 4 and 2 to get 8.

$$22+12\sqrt{2}-\left(4-8\sqrt{2}+8\right)$$

Add 4 and 8 to get 12.

$$22+12\sqrt{2}-\left(12-8\sqrt{2}\right)$$

To find the opposite of 12-8\sqrt{2}, find the opposite of each term.

$$22+12\sqrt{2}-12+8\sqrt{2}$$

Subtract 12 from 22 to get 10.

$$10+12\sqrt{2}+8\sqrt{2}$$

Combine 12\sqrt{2} and 8\sqrt{2} to get 20\sqrt{2}.

$$10+20\sqrt{2}$$