如果实数a,b,c满足a=2b+√2 且ab+√3/2c^2+1/4=0,求a+b+c的值
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如果实数a,b,c满足a=2b+√2 且ab+√3/2c^2+1/4=0,求a+b+c的值
![如果实数a,b,c满足a=2b+√2 且ab+√3/2c^2+1/4=0,求a+b+c的值](/uploads/image/z/17022823-7-3.jpg?t=%E5%A6%82%E6%9E%9C%E5%AE%9E%E6%95%B0a%2Cb%2Cc%E6%BB%A1%E8%B6%B3a%3D2b%2B%E2%88%9A2+%E4%B8%94ab%2B%E2%88%9A3%2F2c%5E2%2B1%2F4%3D0%2C%E6%B1%82a%2Bb%2Bc%E7%9A%84%E5%80%BC)
把a=2b+√2代入
ab+√3/2c^2+1/4=0得
,(2b+√2)b+√3/2c^2+1/4=0,
2b²+√2b+1/4+√3/2c²=0
(√2b)²+2*1/2*√2b+(1/2)²+√3/2c²=0
(√2b+1/2)²+√3/2c²=0
而(√2b+1/2)²≥0,√3/2c²≥0
∴(√2b+1/2=0;√3/2c²=0
解得b= -√2/4; c=0
a=-√2/2+√2=√2/2
∴a+b+c=-√2/4+√2/2+0=√2/4
ab+√3/2c^2+1/4=0得
,(2b+√2)b+√3/2c^2+1/4=0,
2b²+√2b+1/4+√3/2c²=0
(√2b)²+2*1/2*√2b+(1/2)²+√3/2c²=0
(√2b+1/2)²+√3/2c²=0
而(√2b+1/2)²≥0,√3/2c²≥0
∴(√2b+1/2=0;√3/2c²=0
解得b= -√2/4; c=0
a=-√2/2+√2=√2/2
∴a+b+c=-√2/4+√2/2+0=√2/4
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