Solve (sqrt{2}+3/2)^2+(sqrt{2}+1/2)^2

$$( \sqrt { 2 } + \frac { 3 } { 2 } ) ^ { 2 } + ( \sqrt { 2 } + \frac { 1 } { 2 } ) ^ { 2 }$$

Evaluate

$$4\sqrt{2}+\frac{13}{2}\approx 12.156854249$$

Solution Steps

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

$$\left(\sqrt{2}\right)^{2}+3\sqrt{2}+\frac{9}{4}+\left(\sqrt{2}+\frac{1}{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$2+3\sqrt{2}+\frac{9}{4}+\left(\sqrt{2}+\frac{1}{2}\right)^{2}$$

Add 2 and \frac{9}{4} to get \frac{17}{4}.

$$\frac{17}{4}+3\sqrt{2}+\left(\sqrt{2}+\frac{1}{2}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{2}+\frac{1}{2}\right)^{2}.

$$\frac{17}{4}+3\sqrt{2}+\left(\sqrt{2}\right)^{2}+\sqrt{2}+\frac{1}{4}$$

The square of \sqrt{2} is 2.

$$\frac{17}{4}+3\sqrt{2}+2+\sqrt{2}+\frac{1}{4}$$

Add 2 and \frac{1}{4} to get \frac{9}{4}.

$$\frac{17}{4}+3\sqrt{2}+\frac{9}{4}+\sqrt{2}$$

Add \frac{17}{4} and \frac{9}{4} to get \frac{13}{2}.

$$\frac{13}{2}+3\sqrt{2}+\sqrt{2}$$

Combine 3\sqrt{2} and \sqrt{2} to get 4\sqrt{2}.

$$\frac{13}{2}+4\sqrt{2}$$

Expand

$$4 \sqrt{2} + \frac{13}{2} = 12.156854249$$

Solution Steps

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

$$\left(\sqrt{2}\right)^{2}+3\sqrt{2}+\frac{9}{4}+\left(\sqrt{2}+\frac{1}{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$2+3\sqrt{2}+\frac{9}{4}+\left(\sqrt{2}+\frac{1}{2}\right)^{2}$$

Add 2 and \frac{9}{4} to get \frac{17}{4}.

$$\frac{17}{4}+3\sqrt{2}+\left(\sqrt{2}+\frac{1}{2}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{2}+\frac{1}{2}\right)^{2}.

$$\frac{17}{4}+3\sqrt{2}+\left(\sqrt{2}\right)^{2}+\sqrt{2}+\frac{1}{4}$$

The square of \sqrt{2} is 2.

$$\frac{17}{4}+3\sqrt{2}+2+\sqrt{2}+\frac{1}{4}$$

Add 2 and \frac{1}{4} to get \frac{9}{4}.

$$\frac{17}{4}+3\sqrt{2}+\frac{9}{4}+\sqrt{2}$$

Add \frac{17}{4} and \frac{9}{4} to get \frac{13}{2}.

$$\frac{13}{2}+3\sqrt{2}+\sqrt{2}$$

Combine 3\sqrt{2} and \sqrt{2} to get 4\sqrt{2}.

$$\frac{13}{2}+4\sqrt{2}$$