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

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

Evaluate

$$4x^{2}-20x+50$$

Solution Steps

Expand \left(\sqrt{2}x\right)^{2}.

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

The square of \sqrt{2} is 2.

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

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

$$2x^{2}+25\left(\sqrt{2}\right)^{2}-10\sqrt{2}\sqrt{2}x+\left(\sqrt{2}\right)^{2}x^{2}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$2x^{2}+25\left(\sqrt{2}\right)^{2}-10\times 2x+\left(\sqrt{2}\right)^{2}x^{2}$$

The square of \sqrt{2} is 2.

$$2x^{2}+25\times 2-10\times 2x+\left(\sqrt{2}\right)^{2}x^{2}$$

Multiply 25 and 2 to get 50.

$$2x^{2}+50-10\times 2x+\left(\sqrt{2}\right)^{2}x^{2}$$

Multiply -10 and 2 to get -20.

$$2x^{2}+50-20x+\left(\sqrt{2}\right)^{2}x^{2}$$

The square of \sqrt{2} is 2.

$$2x^{2}+50-20x+2x^{2}$$

Combine 2x^{2} and 2x^{2} to get 4x^{2}.

$$4x^{2}+50-20x$$

Expand

$$4x^{2}-20x+50$$

Solution Steps

Expand \left(\sqrt{2}x\right)^{2}.

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

The square of \sqrt{2} is 2.

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

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

$$2x^{2}+25\left(\sqrt{2}\right)^{2}-10\sqrt{2}\sqrt{2}x+\left(\sqrt{2}\right)^{2}x^{2}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$2x^{2}+25\left(\sqrt{2}\right)^{2}-10\times 2x+\left(\sqrt{2}\right)^{2}x^{2}$$

The square of \sqrt{2} is 2.

$$2x^{2}+25\times 2-10\times 2x+\left(\sqrt{2}\right)^{2}x^{2}$$

Multiply 25 and 2 to get 50.

$$2x^{2}+50-10\times 2x+\left(\sqrt{2}\right)^{2}x^{2}$$

Multiply -10 and 2 to get -20.

$$2x^{2}+50-20x+\left(\sqrt{2}\right)^{2}x^{2}$$

The square of \sqrt{2} is 2.

$$2x^{2}+50-20x+2x^{2}$$

Combine 2x^{2} and 2x^{2} to get 4x^{2}.

$$4x^{2}+50-20x$$