Question 1

Algebra Level 3

Evaluate: a 2 + a × b − b 2 a^{2}+a \times b-b^{2} Given: a= 2 + 1 \sqrt{2}+1 , b= 3 − 1 \sqrt{3}-1

Note: Round your answer to the thousandth place.


The answer is 7.060.

Luye Xue by Luye Xue , 19, China

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1 solution

Luye Xue
Feb 11, 2015

a 2 + a b − b 2 a^{2}+ab-b^{2}

= ( 2 + 1 ) 2 + ( 2 + 1 ) ( 3 − 1 ) − ( 3 − 1 ) 2 (\sqrt{2}+1) ^{2}+(\sqrt{2}+1)(\sqrt{3}-1)-(\sqrt{3}-1)^{2}

= ( 2 + 2 2 + 1 ) + ( 6 − 2 ) + ( 3 − 1 ) − ( 3 − 2 3 + 1 ) (2+2\sqrt{2}+1)+(\sqrt{6}-\sqrt{2})+(\sqrt{3}-1)-(3-2\sqrt{3}+1)

= 2 + 2 2 + 1 + 6 − 2 + 3 − 1 − 3 + 2 3 − 1 2+2\sqrt{2}+1+\sqrt{6}-\sqrt{2}+\sqrt{3}-1-3+2\sqrt{3}-1

= − 2 + 2 + 3 3 + 6 -2+\sqrt{2}+3\sqrt{3}+\sqrt{6}

nearest thousandth place: 7.060

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