Solve {left(150-475sqrt{2}right)}^2+{left(750-475sqrt{2}right)}^2

$${ \left(150-475 \sqrt{ 2 } \right) }^{ 2 } + { \left(750-475 \sqrt{ 2 } \right) }^{ 2 }$$

Evaluate

$$1487500-855000\sqrt{2}\approx 278347.404171004$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(150-475\sqrt{2}\right)^{2}.

$$22500-142500\sqrt{2}+225625\left(\sqrt{2}\right)^{2}+\left(750-475\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$22500-142500\sqrt{2}+225625\times 2+\left(750-475\sqrt{2}\right)^{2}$$

Multiply 225625 and 2 to get 451250.

$$22500-142500\sqrt{2}+451250+\left(750-475\sqrt{2}\right)^{2}$$

Add 22500 and 451250 to get 473750.

$$473750-142500\sqrt{2}+\left(750-475\sqrt{2}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(750-475\sqrt{2}\right)^{2}.

$$473750-142500\sqrt{2}[phone]\sqrt{2}+225625\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$473750-142500\sqrt{2}[phone]\sqrt{2}+225625\times 2$$

Multiply 225625 and 2 to get 451250.

$$473750-142500\sqrt{2}[phone]\sqrt{2}+451250$$

Add 562500 and 451250 to get 1013750.

$$473750-142500\sqrt{2}[phone]\sqrt{2}$$

Add 473750 and 1013750 to get 1487500.

$$1487500-142500\sqrt{2}-712500\sqrt{2}$$

Combine -142500\sqrt{2} and -712500\sqrt{2} to get -855000\sqrt{2}.

$$1487500-855000\sqrt{2}$$

Expand

$$1487500-855000\sqrt{2}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(150-475\sqrt{2}\right)^{2}.

$$22500-142500\sqrt{2}+225625\left(\sqrt{2}\right)^{2}+\left(750-475\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$22500-142500\sqrt{2}+225625\times 2+\left(750-475\sqrt{2}\right)^{2}$$

Multiply 225625 and 2 to get 451250.

$$22500-142500\sqrt{2}+451250+\left(750-475\sqrt{2}\right)^{2}$$

Add 22500 and 451250 to get 473750.

$$473750-142500\sqrt{2}+\left(750-475\sqrt{2}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(750-475\sqrt{2}\right)^{2}.

$$473750-142500\sqrt{2}[phone]\sqrt{2}+225625\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$473750-142500\sqrt{2}[phone]\sqrt{2}+225625\times 2$$

Multiply 225625 and 2 to get 451250.

$$473750-142500\sqrt{2}[phone]\sqrt{2}+451250$$

Add 562500 and 451250 to get 1013750.

$$473750-142500\sqrt{2}[phone]\sqrt{2}$$

Add 473750 and 1013750 to get 1487500.

$$1487500-142500\sqrt{2}-712500\sqrt{2}$$

Combine -142500\sqrt{2} and -712500\sqrt{2} to get -855000\sqrt{2}.

$$1487500-855000\sqrt{2}$$