Question

chứng minh các bthuc sau luôn có giá trị dương với mọi biến E=x^2+(x+1)^2

 

Answer 1

\(E=x^2+\left(x+1\right)^2\)

\(=x^2+x^2+2x+1\)

\(=2x^2+2x+1\)

\(=2\left(x^2+x+\dfrac{1}{2}\right)\)

\(=2\left(x^2+x+\dfrac{1}{4}+\dfrac{1}{4}\right)\)

\(=2\left(x+\dfrac{1}{2}\right)^2+\dfrac{1}{2}>=\dfrac{1}{2}>0\forall x\)

Answer 2

`E = x^2 + (x+1)^2 `

`= x^2 + x^2 + 2x + 1`

`= 2x^2 + 2x + 1`

`= 2(x^2 + x + 1/2) `

`= 2(x^2 + 2 . 1x . 1/2 + 1/4 + 1/4) `

`= 2(x^2 + 2 . 1x . 1/2 + 1/4) + 1/2`

`= 2 (x - 1/2)^2 + 1/2`

Do `(x - 1/2)^2 >= 0 ∀ x`

`<=> 2 (x - 1/2)^2  >= 0 ∀ x`

`<=> 2 (x - 1/2)^2 + 1/2 >=1/2 ∀ x`

Hay `E > 0 ∀ x (đpcm)`