Solve {left(7+3sqrt{2}right)}^2-{left(5-2sqrt{5}right)}^2

$${ \left(7+3 \sqrt{ 2 } \right) }^{ 2 } - { \left(5-2 \sqrt{ 5 } \right) }^{ 2 }$$

Evaluate

$$20\sqrt{5}+42\sqrt{2}+22\approx 126.11832917$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(7+3\sqrt{2}\right)^{2}.

$$49+42\sqrt{2}+9\left(\sqrt{2}\right)^{2}-\left(5-2\sqrt{5}\right)^{2}$$

The square of \sqrt{2} is 2.

$$49+42\sqrt{2}+9\times 2-\left(5-2\sqrt{5}\right)^{2}$$

Multiply 9 and 2 to get 18.

$$49+42\sqrt{2}+18-\left(5-2\sqrt{5}\right)^{2}$$

Add 49 and 18 to get 67.

$$67+42\sqrt{2}-\left(5-2\sqrt{5}\right)^{2}$$

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

$$67+42\sqrt{2}-\left(25-20\sqrt{5}+4\left(\sqrt{5}\right)^{2}\right)$$

The square of \sqrt{5} is 5.

$$67+42\sqrt{2}-\left(25-20\sqrt{5}+4\times 5\right)$$

Multiply 4 and 5 to get 20.

$$67+42\sqrt{2}-\left(25-20\sqrt{5}+20\right)$$

Add 25 and 20 to get 45.

$$67+42\sqrt{2}-\left(45-20\sqrt{5}\right)$$

To find the opposite of 45-20\sqrt{5}, find the opposite of each term.

$$67+42\sqrt{2}-45+20\sqrt{5}$$

Subtract 45 from 67 to get 22.

$$22+42\sqrt{2}+20\sqrt{5}$$

Expand

$$20 \sqrt{5} + 42 \sqrt{2} + 22 = 126.11832917$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(7+3\sqrt{2}\right)^{2}.

$$49+42\sqrt{2}+9\left(\sqrt{2}\right)^{2}-\left(5-2\sqrt{5}\right)^{2}$$

The square of \sqrt{2} is 2.

$$49+42\sqrt{2}+9\times 2-\left(5-2\sqrt{5}\right)^{2}$$

Multiply 9 and 2 to get 18.

$$49+42\sqrt{2}+18-\left(5-2\sqrt{5}\right)^{2}$$

Add 49 and 18 to get 67.

$$67+42\sqrt{2}-\left(5-2\sqrt{5}\right)^{2}$$

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

$$67+42\sqrt{2}-\left(25-20\sqrt{5}+4\left(\sqrt{5}\right)^{2}\right)$$

The square of \sqrt{5} is 5.

$$67+42\sqrt{2}-\left(25-20\sqrt{5}+4\times 5\right)$$

Multiply 4 and 5 to get 20.

$$67+42\sqrt{2}-\left(25-20\sqrt{5}+20\right)$$

Add 25 and 20 to get 45.

$$67+42\sqrt{2}-\left(45-20\sqrt{5}\right)$$

To find the opposite of 45-20\sqrt{5}, find the opposite of each term.

$$67+42\sqrt{2}-45+20\sqrt{5}$$

Subtract 45 from 67 to get 22.

$$22+42\sqrt{2}+20\sqrt{5}$$