Question

1, x²- 8x + 20

2,4x²- 12x + 11

3,x² - 2x + y²+4y + 6

4, (15x-1)²+3(7x+3)(x+1)-(x²-73)
chứng minh các biểu thức sau đây không âm với mọi x,y

giúp mik nhé:))

Answer 1

1) \(x^2-8x+20=\left(x^2-8x+16\right)+4=\left(x-4\right)^2+4>0\forall x\)

(do \(\left(x-4\right)^2\ge0\forall x\)

2) \(4x^2-12x+11=\left(4x^2-12x+9\right)+2=\left(2x-3\right)^2+2>0\forall x\)

(do \(\left(2x-3\right)^2\ge0\forall x\))

3) \(x^2-2x+y^2+4y+6=\left(x-1\right)^2+\left(y+2\right)^2+1>0\forall x;y\)

(do ....)

4) \(\left(15x-1\right)^2+3\left(7x+3\right)\left(x+1\right)-\left(x^2-73\right)\)

\(=225x^2-30x+1+3\left(7x^2+10x+3\right)-x^2+73\)

\(=225x^2-30x+83+21x^2+30x-x^2\)

\(=245x^2+83>0\forall x\)