Question

chứng minh biểu thức luôn dương với a

a)x⁴-x²+3b)x²-x+1c)x²+x+2d)(x+3)(x-11)+20

Answer 1

a) \(x^4-x^2+3=\left[\left(x^2\right)^2-2\cdot x^2\cdot\frac{1}{2}+\frac{1}{4}\right]+\frac{11}{4}=\left(x^2-\frac{1}{2}\right)^2+\frac{11}{4}>0\)

=>đpcm

b) \(x^2-x+1=\left(x^2-2\cdot x\cdot\frac{1}{2}+\frac{1}{4}\right)+\frac{3}{4}=\left(x-\frac{1}{2}\right)^2+\frac{3}{4}>0\)

=>đpcm

c) \(x^2+x+2=\left(x^2+2\cdot x+\frac{1}{2}+\frac{1}{4}\right)+\frac{7}{4}=\left(x+\frac{1}{2}\right)^2+\frac{7}{4}>0\)

=>đpcm

d) \(\left(x+3\right)\left(x-11\right)+20\)

\(=x^2-11x+3x-33+20\)

\(=x^2-8x-13\)

\(=\left(x^2-8x+16\right)-29=\left(x+4\right)^2-29\)

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