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

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

Evaluate

$$18-14\sqrt{2}\approx -1.798989873$$

Solution Steps

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

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

The square of \sqrt{2} is 2.

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

Add 1 and 2 to get 3.

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

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

$$3-2\sqrt{2}+4\left(\sqrt{2}\right)^{2}-12\sqrt{2}+9-\left(3\sqrt{2}-4\right)\left(4+3\sqrt{2}\right)$$

The square of \sqrt{2} is 2.

$$3-2\sqrt{2}+4\times 2-12\sqrt{2}+9-\left(3\sqrt{2}-4\right)\left(4+3\sqrt{2}\right)$$

Multiply 4 and 2 to get 8.

$$3-2\sqrt{2}+8-12\sqrt{2}+9-\left(3\sqrt{2}-4\right)\left(4+3\sqrt{2}\right)$$

Add 8 and 9 to get 17.

$$3-2\sqrt{2}+17-12\sqrt{2}-\left(3\sqrt{2}-4\right)\left(4+3\sqrt{2}\right)$$

Add 3 and 17 to get 20.

$$20-2\sqrt{2}-12\sqrt{2}-\left(3\sqrt{2}-4\right)\left(4+3\sqrt{2}\right)$$

Combine -2\sqrt{2} and -12\sqrt{2} to get -14\sqrt{2}.

$$20-14\sqrt{2}-\left(3\sqrt{2}-4\right)\left(4+3\sqrt{2}\right)$$

Use the distributive property to multiply 3\sqrt{2}-4 by 4+3\sqrt{2} and combine like terms.

$$20-14\sqrt{2}-\left(9\left(\sqrt{2}\right)^{2}-16\right)$$

The square of \sqrt{2} is 2.

$$20-14\sqrt{2}-\left(9\times 2-16\right)$$

Multiply 9 and 2 to get 18.

$$20-14\sqrt{2}-\left(18-16\right)$$

Subtract 16 from 18 to get 2.

$$20-14\sqrt{2}-2$$

Subtract 2 from 20 to get 18.

$$18-14\sqrt{2}$$