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已知實數\(a,b,c\)滿足條件\(a=\sqrt{2}+b\)，且\(2ab+2\sqrt{2}c^2+1=0\)，試問下列何者正確？
(A)\(\displaystyle a=\frac{-\sqrt{2}}{2}\) (B)\(\displaystyle a=\frac{\sqrt{2}}{2}\) (C)\(\displaystyle b=-\frac{\sqrt{2}}{2}\) (D)\(\displaystyle b=\frac{\sqrt{2}}{2}\) (E)\(a+b+c=0\)
[解答]
將\(a=\sqrt{2}+b\)代入 \(2ab+2\sqrt{2}c^2+1=0\)
得到\(2(b^2+\sqrt{2}b+\frac{1}{2})+2\sqrt{2}c^2=0\)
在配方得到\(2(b+\frac{\sqrt{2}}{2})^2+2\sqrt{2}c^2=0\)
所以\(b=-\frac{\sqrt{2}}{2},c=0\)
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