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

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

Evaluate

$$76$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3\sqrt{2}+2\sqrt{5}\right)^{2}.

$$9\left(\sqrt{2}\right)^{2}+12\sqrt{2}\sqrt{5}+4\left(\sqrt{5}\right)^{2}+\left(3\sqrt{2}-2\sqrt{5}\right)^{2}$$

The square of \sqrt{2} is 2.

$$9\times 2+12\sqrt{2}\sqrt{5}+4\left(\sqrt{5}\right)^{2}+\left(3\sqrt{2}-2\sqrt{5}\right)^{2}$$

Multiply 9 and 2 to get 18.

$$18+12\sqrt{2}\sqrt{5}+4\left(\sqrt{5}\right)^{2}+\left(3\sqrt{2}-2\sqrt{5}\right)^{2}$$

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

$$18+12\sqrt{10}+4\left(\sqrt{5}\right)^{2}+\left(3\sqrt{2}-2\sqrt{5}\right)^{2}$$

The square of \sqrt{5} is 5.

$$18+12\sqrt{10}+4\times 5+\left(3\sqrt{2}-2\sqrt{5}\right)^{2}$$

Multiply 4 and 5 to get 20.

$$18+12\sqrt{10}+20+\left(3\sqrt{2}-2\sqrt{5}\right)^{2}$$

Add 18 and 20 to get 38.

$$38+12\sqrt{10}+\left(3\sqrt{2}-2\sqrt{5}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3\sqrt{2}-2\sqrt{5}\right)^{2}.

$$38+12\sqrt{10}+9\left(\sqrt{2}\right)^{2}-12\sqrt{2}\sqrt{5}+4\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{2} is 2.

$$38+12\sqrt{10}+9\times 2-12\sqrt{2}\sqrt{5}+4\left(\sqrt{5}\right)^{2}$$

Multiply 9 and 2 to get 18.

$$38+12\sqrt{10}+18-12\sqrt{2}\sqrt{5}+4\left(\sqrt{5}\right)^{2}$$

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

$$38+12\sqrt{10}+18-12\sqrt{10}+4\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{5} is 5.

$$38+12\sqrt{10}+18-12\sqrt{10}+4\times 5$$

Multiply 4 and 5 to get 20.

$$38+12\sqrt{10}+18-12\sqrt{10}+20$$

Add 18 and 20 to get 38.

$$38+12\sqrt{10}+38-12\sqrt{10}$$

Add 38 and 38 to get 76.

$$76+12\sqrt{10}-12\sqrt{10}$$

Combine 12\sqrt{10} and -12\sqrt{10} to get 0.

$$76$$

Factor

$$2^{2}\times 19$$