Solve sqrt{1^2+2^2}+sqrt{2^2+4^2}+x

$$\sqrt{ { 1 }^{ 2 } + { 2 }^{ 2 } } + \sqrt{ { 2 }^{ 2 } + { 4 }^{ 2 } } +x$$

Evaluate

$$x+3\sqrt{5}$$

Solution Steps

Calculate 1 to the power of 2 and get 1.

$$\sqrt{1+2^{2}}+\sqrt{2^{2}+4^{2}}+x$$

Calculate 2 to the power of 2 and get 4.

$$\sqrt{1+4}+\sqrt{2^{2}+4^{2}}+x$$

Add 1 and 4 to get 5.

$$\sqrt{5}+\sqrt{2^{2}+4^{2}}+x$$

Calculate 2 to the power of 2 and get 4.

$$\sqrt{5}+\sqrt{4+4^{2}}+x$$

Calculate 4 to the power of 2 and get 16.

$$\sqrt{5}+\sqrt{4+16}+x$$

Add 4 and 16 to get 20.

$$\sqrt{5}+\sqrt{20}+x$$

Factor 20=2^{2}\times 5. Rewrite the square root of the product \sqrt{2^{2}\times 5} as the product of square roots \sqrt{2^{2}}\sqrt{5}. Take the square root of 2^{2}.

$$\sqrt{5}+2\sqrt{5}+x$$

Combine \sqrt{5} and 2\sqrt{5} to get 3\sqrt{5}.

$$3\sqrt{5}+x$$

Differentiate w.r.t. x

$$1$$