Solve 1/2*(5+2sqrt{2})(5-2sqrt{2})

$$\frac { 1 } { 2 } \times ( 5 + 2 \sqrt { 2 } ) ( 5 - 2 \sqrt { 2 } )$$

Evaluate

$$\frac{17}{2}=8.5$$

Solution Steps

Use the distributive property to multiply \frac{1}{2} by 5+2\sqrt{2}.

$$\left(\frac{1}{2}\times 5+\frac{1}{2}\times 2\sqrt{2}\right)\left(5-2\sqrt{2}\right)$$

Multiply \frac{1}{2} and 5 to get \frac{5}{2}.

$$\left(\frac{5}{2}+\frac{1}{2}\times 2\sqrt{2}\right)\left(5-2\sqrt{2}\right)$$

Cancel out 2 and 2.

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

Apply the distributive property by multiplying each term of \frac{5}{2}+\sqrt{2} by each term of 5-2\sqrt{2}.

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

Express \frac{5}{2}\times 5 as a single fraction.

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

Multiply 5 and 5 to get 25.

$$\frac{25}{2}+\frac{5}{2}\left(-2\right)\sqrt{2}+5\sqrt{2}-2\left(\sqrt{2}\right)^{2}$$

Express \frac{5}{2}\left(-2\right) as a single fraction.

$$\frac{25}{2}+\frac{5\left(-2\right)}{2}\sqrt{2}+5\sqrt{2}-2\left(\sqrt{2}\right)^{2}$$

Multiply 5 and -2 to get -10.

$$\frac{25}{2}+\frac{-10}{2}\sqrt{2}+5\sqrt{2}-2\left(\sqrt{2}\right)^{2}$$

Divide -10 by 2 to get -5.

$$\frac{25}{2}-5\sqrt{2}+5\sqrt{2}-2\left(\sqrt{2}\right)^{2}$$

Combine -5\sqrt{2} and 5\sqrt{2} to get 0.

$$\frac{25}{2}-2\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$\frac{25}{2}-2\times 2$$

Multiply -2 and 2 to get -4.

$$\frac{25}{2}-4$$

Convert 4 to fraction \frac{8}{2}.

$$\frac{25}{2}-\frac{8}{2}$$

Since \frac{25}{2} and \frac{8}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{25-8}{2}$$

Subtract 8 from 25 to get 17.

$$\frac{17}{2}$$

Factor

$$\frac{17}{2} = 8\frac{1}{2} = 8.5$$