Solve (3-52)*(5+sqrt{2})-(sqrt{2}-1)^2

$$( 3 - 52 ) \cdot ( 5 + \sqrt { 2 } ) - ( \sqrt { 2 } - 1 ) ^ { 2 }$$

Evaluate

$$-47\sqrt{2}-248\approx -314.468037432$$

Solution Steps

Subtract 52 from 3 to get -49.

$$-49\left(5+\sqrt{2}\right)-\left(\sqrt{2}-1\right)^{2}$$

Use the distributive property to multiply -49 by 5+\sqrt{2}.

$$-245-49\sqrt{2}-\left(\sqrt{2}-1\right)^{2}$$

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

$$-245-49\sqrt{2}-\left(\left(\sqrt{2}\right)^{2}-2\sqrt{2}+1\right)$$

The square of \sqrt{2} is 2.

$$-245-49\sqrt{2}-\left(2-2\sqrt{2}+1\right)$$

Add 2 and 1 to get 3.

$$-245-49\sqrt{2}-\left(3-2\sqrt{2}\right)$$

To find the opposite of 3-2\sqrt{2}, find the opposite of each term.

$$-245-49\sqrt{2}-3+2\sqrt{2}$$

Subtract 3 from -245 to get -248.

$$-248-49\sqrt{2}+2\sqrt{2}$$

Combine -49\sqrt{2} and 2\sqrt{2} to get -47\sqrt{2}.

$$-248-47\sqrt{2}$$