Bài 13:

\(1)A=x^2-x+1\\ =\left(x^2-2\cdot x\cdot\dfrac{1}{2}+\dfrac{1}{4}\right)+\dfrac{3}{4}\\ =\left(x-\dfrac{1}{2}\right)^2+\dfrac{3}{4}\ge\dfrac{3}{4}>0\forall x\\ 2)B=x^2+x+1\\ =\left(x^2+2\cdot x\cdot\dfrac{1}{2}+\dfrac{1}{4}\right)+\dfrac{3}{4}\\ =\left(x+\dfrac{1}{2}\right)^2+\dfrac{3}{4}\ge\dfrac{3}{4}>0\forall x\\ 3)C=x^2+2x+2\\ =\left(x^2+2x+1\right)+1\\ =\left(x+1\right)^2+1\ge1>0\forall x\)

\(4)A=x^2-5x+10\\ =\left(x^2-2\cdot x\cdot\dfrac{5}{2}+\dfrac{25}{4}\right)+\dfrac{15}{4}\\ =\left(x-\dfrac{5}{2}\right)^2+\dfrac{15}{4}\ge\dfrac{15}{4}>0\forall x\\ 5)B=x^2-8x+20\\ =\left(x^2-8x+16\right)+4\\ =\left(x-4\right)^2+4\ge4>0\forall x\\ 6)C=x^2-8x+17\\ =\left(x^2-8x+16\right)+1\\ =\left(x-4\right)^2+1\ge1>0\forall x\) 

\(7)A=x^2-6x+10\\ =\left(x^2-6x+9\right)+1\\ =\left(x-3\right)^2+1\ge1>0\forall x\\ 8)B=9x^2-6x+2\\ =\left(9x^2-6x+1\right)+1\\ =\left(3x-1\right)^2+1\ge1>0\forall x\\ 9)C=2x^2+8x+15\\ =\left(2x^2+8x+8\right)+7\\ =2\left(x^2+4x+4\right)+7\\ =2\left(x+2\right)^2+7\ge7>0\forall x\)

\(a)\left(x+5\right)^2=x^2+2\cdot x\cdot5+5^2=x^2+10x+25\\ b)\left(2-y\right)^2=2^2-2\cdot2\cdot y+y^2=4-4y+y^2\\ c)\left(5x-1\right)^2=\left(5x\right)^2-2\cdot5x\cdot1+1^2=25x^2-10x+1\\ d)\left(1+5x^3\right)^2=1^2+2\cdot1\cdot5x^3+\left(5x^3\right)^2=1+10x^3+25x^6\\ e)\left(7-a^2\right)\left(7+a^2\right)=7^2-\left(a^2\right)^2=49-a^4\\ \left(x-y\right)^2-\left(x+y\right)^2=\left(x-y+x+y\right)\left(x-y-x-y\right)=2x\cdot-2y=-4xy\\ g)\left(2x^3-\dfrac{1}{2}y\right)^2=\left(2x^3\right)^2-2\cdot2x^3\cdot\dfrac{1}{2}y+\left(\dfrac{1}{2}y^2\right)=4x^6-2x^3y+\dfrac{1}{4}y^2\\ h)\left(x^2+4y\right)^2=\left(x^2\right)^2+2\cdot x^2\cdot4y+\left(4y\right)^2=x^4+8x^2y+16y^2\\ i)\left(a+b+c\right)^2=\left[a+\left(b+c\right)\right]^2=a^2+2a\left(b+c\right)+\left(b+c\right)^2\\ =a^2+2ab+2ac+b^2+2bc+c^2=a^2+b^2+c^2+2ab+2bc+2ac\\ k)\left(a-b-c\right)^2=\left[a-\left(b+c\right)\right]^2=a^2-2a\left(b+c\right)+\left(b+c\right)^2\\ =a^2-2ab-2ac+b^2+2bc+c^2=a^2+b^2+c^2-2ab-2ac+2bc\)

Bài 2:

\(a)2x^2y-\dfrac{1}{4}x^2y+5x^2y-4x^2y\\ =x^2y\cdot\left(2-\dfrac{1}{4}+5-4\right)\\ =x^2y\cdot\left(3-\dfrac{1}{4}\right)\\ =\dfrac{11}{4}x^2y\\ b)5y^3z^2-3y^3z^2+7y^3z^2-6y^3z^2\\ =y^3z^2\cdot\left(5-3+7-6\right)\\ =3y^3z^2\\ c)-4x^3y^4+6x^2y^3+\dfrac{1}{2}x^3y^4-\dfrac{3}{2}x^2y^3\\ =\left(\dfrac{1}{2}x^3y^4-4x^3y^4\right)+\left(6x^2y^3-\dfrac{3}{2}x^2y^3\right)\\ =x^3y^4\left(\dfrac{1}{2}-4\right)+x^2y^3\left(6-\dfrac{3}{2}\right)\\ =-\dfrac{7}{2}x^3y^4+\dfrac{9}{2}x^2y^3\)

Bài 8:

1: \(\left(x+y\right)^2-\left(x-y\right)^2\)

\(=\left(x+y+x-y\right)\left(x+y-x+y\right)\)

\(=2x\cdot2y=4xy\)

2: \(\left(2x+3\right)^2-3x\left(2x+1\right)\)

\(=4x^2+12x+9-6x^2-3x\)

\(=-2x^2+9x+9\)

3: \(\left(4-2x\right)\left(4+2x\right)-4x\left(2x+3\right)\)

\(=4^2-\left(2x\right)^2-8x^2-12x\)

\(=16-4x^2-8x^2-12x=-12x^2-12x+16\)

4: \(2\left(x+y\right)\left(x-y\right)+\left(x+y\right)^2-2x^2\)

\(=2\left(x^2-y^2\right)+x^2+2xy+y^2-2x^2\)

\(=2x^2-2y^2-x^2+2xy+y^2=x^2+2xy-y^2\)

5: \(\left(3x+4\right)\left(3x-2\right)-\left(3x+1\right)^2\)

\(=9x^2-6x+12x-8-9x^2-6x-1\)

=-9

6: \(4x\left(x-3\right)-\left(2x-1\right)\left(2x+1\right)\)

\(=4x^2-12x-\left(4x^2-1\right)\)

\(=4x^2-12x-4x^2+1=-12x+1\)

7: \(\dfrac{3}{2}x^2-\left(x-1\right)\left(x+1\right)+3x\)

\(=\dfrac{3}{2}x^2+3x-\left(x^2-1\right)\)

\(=\dfrac{3}{2}x^2+3x-x^2+1=\dfrac{1}{2}x^2+3x+1\)

8: \(2\left(5-x\right)\left(5+x\right)-\left(2x+3\right)^2-x\left(3x+2\right)\)

\(=2\left(25-x^2\right)-4x^2-12x-9-3x^2-2x\)

\(=2\left(25-x^2\right)-7x^2-14x-9\)

\(=50-2x^2-7x^2-14x-9=-9x^2-14x+41\)

Giúp tớ nhanh vs ạ

Bài 8:

\(1)\left(x+y\right)^2-\left(x-y\right)^2\\ =\left(x^2+2xy+y^2\right)-\left(x^2-2xy+y^2\right)\\ =x^2+2xy+y^2-x^2+2xy-y^2\\ =4xy\\ 2)\left(2x+3\right)^2-3x\left(2x+1\right)\\ =\left(4x^2+12x+9\right)-\left(6x^2+3x\right)\\ =4x^2+12x+9-6x^2-3x\\ =-2x^2+9x+9\\ 3)\left(4-2x\right)\left(4+2x\right)-4x\left(2x+3\right)\\ =\left[4^2-\left(2x\right)^2\right]-\left(8x^2+12x\right)\\ =16-4x^2-8x^2-12x\\ =16-12x^2-12x\\ 4)2\left(x+y\right)\left(x-y\right)+\left(x+y\right)^2-2x^2\\ =2\left(x^2-y^2\right)+\left(x^2+2xy+y^2\right)-2x^2\\ =2x^2-2y^2+x^2+2xy+y^2-2x^2\\ =x^2+2xy-y^2\)

Bài 8:

1: \(\left(x+y\right)^2-\left(x-y\right)^2\)

\(=\left(x+y+x-y\right)\left(x+y-x+y\right)\)

\(=2x\cdot2y=4xy\)

2: \(\left(2x+3\right)^2-3x\left(2x+1\right)\)

\(=4x^2+12x+9-6x^2-3x\)

\(=-2x^2+9x+9\)

3: \(\left(4-2x\right)\left(4+2x\right)-4x\left(2x+3\right)\)

\(=4^2-\left(2x\right)^2-8x^2-12x\)

\(=16-4x^2-8x^2-12x=-12x^2-12x+16\)

4: \(2\left(x+y\right)\left(x-y\right)+\left(x+y\right)^2-2x^2\)

\(=2\left(x^2-y^2\right)+x^2+2xy+y^2-2x^2\)

\(=2x^2-2y^2-x^2+2xy+y^2=x^2+2xy-y^2\)

5: \(\left(3x+4\right)\left(3x-2\right)-\left(3x+1\right)^2\)

\(=9x^2-6x+12x-8-9x^2-6x-1\)

=-9

6: \(4x\left(x-3\right)-\left(2x-1\right)\left(2x+1\right)\)

\(=4x^2-12x-\left(4x^2-1\right)\)

\(=4x^2-12x-4x^2+1=-12x+1\)

7: \(\dfrac{3}{2}x^2-\left(x-1\right)\left(x+1\right)+3x\)

\(=\dfrac{3}{2}x^2+3x-\left(x^2-1\right)\)

\(=\dfrac{3}{2}x^2+3x-x^2+1=\dfrac{1}{2}x^2+3x+1\)

8: \(2\left(5-x\right)\left(5+x\right)-\left(2x+3\right)^2-x\left(3x+2\right)\)

\(=2\left(25-x^2\right)-4x^2-12x-9-3x^2-2x\)

\(=2\left(25-x^2\right)-7x^2-14x-9\)

\(=50-2x^2-7x^2-14x-9=-9x^2-14x+41\)

11.

a) 

\(A=\left(x+1\right)\left(x^2-x+1\right)-\left(x-1\right)\left(x^2+x+1\right)\\=\left(x+1\right)\left(x^2-x\cdot1+1^2\right)-\left(x-1\right)\left(x^2+x\cdot1+1^2\right)\\ =\left(x^3+1^3\right)-\left(x^3-1^3\right)\\ =x^3+1-x^3+1\\ =2\)

=> Giá trị của bt không phụ thuộc vào biến 

b) 

\(B=\left(2x+6\right)\left(4x^2-12x+36\right)-8x^3+10\\ =\left(2x+6\right)\left[\left(2x\right)^2-2x\cdot6+6^2\right]-8x^3+10\\ =\left[\left(2x\right)^3+6^3\right]-8x^3+10\\ =\left(8x^3+216\right)-8x^3+10\\ =8x^3+216-8x^3+10\\ =226\)

=> Giá trị của bt không phụ thuộc vào biến

6.

\(a)\left(x+1\right)^3=x^3+3\cdot x^2\cdot1+3\cdot x\cdot1^2+1^3=x^3+3x^2+3x+1\\ b)\left(2x+3\right)^3=\left(2x\right)^3+3\cdot\left(2x\right)^2\cdot3+3\cdot2x\cdot3^2+3^3=8x^3+36x^2+54x+27\\ c)\left(x^2+2\right)^3=\left(x^2\right)^3+3\cdot\left(x^2\right)^2\cdot2+3\cdot x^2\cdot2^2+2^3=x^6+6x^4+12x^2+8\\ d)\left(2x+5y\right)^3=\left(2x\right)^3+3\cdot\left(2x\right)^2\cdot5y+3\cdot2x\cdot\left(5y\right)^2+\left(5y\right)^3=8x^3+60x^2y+150xy^2+125y^3\\ e.\left(x+\dfrac{1}{2}\right)^3=x^3+3\cdot x^2\cdot\dfrac{1}{2}+3\cdot x\cdot\left(\dfrac{1}{2}\right)^2+\left(\dfrac{1}{2}\right)^3=x^3+\dfrac{3}{2}x^2+\dfrac{3}{4}x+\dfrac{1}{8}\\ g.\left(\dfrac{1}{2}x+y^2\right)=\left(\dfrac{1}{2}x\right)^3+3\cdot\left(\dfrac{1}{2}x\right)^2\cdot y^2+3\cdot\dfrac{1}{2}x\cdot\left(y^2\right)^2+\left(y^2\right)^3\\ =\dfrac{x^3}{8}+\dfrac{3}{4}x^2y^2+\dfrac{3}{2}xy^4+y^6\\ h.\left(x^2-2\right)^3=\left(x^2\right)^3-3\cdot\left(x^2\right)^2\cdot2+3\cdot x^2\cdot2^2-2^3=x^6-6x^4+12x^2-8\)

\(d.x^{11}+x^7+1\\ =x^{11}-x^2+x^7-x+x^2+x+1\\ =x^2\left(x^9-1\right)+x\left(x^6-1\right)+\left(x^2+x+1\right)\\ =x^2\left(x^3-1\right)\left(x^6+x^3+1\right)+x\left(x^3-1\right)\left(x^3+1\right)+\left(x^2+x+1\right)\\ =x^2\left(x-1\right)\left(x^2+x+1\right)\left(x^6+x^3+1\right)+x\left(x-1\right)\left(x^2+x+1\right)\left(x^3+1\right)+\left(x^2+x+1\right)\\=\left(x^2+x+1\right)\left[x^2\left(x-1\right)\left(x^6+x^3+1\right)+x\left(x-1\right)\left(x^3+1\right)+1\right]\\ =\left(x^2+x+1\right)\left[\left(x^3-x^2\right)\left(x^6+x^3+1\right)+\left(x^2-x\right)\left(x^3+1\right)+1\right]\\ =\left(x^2+x+1\right)\left(x^9+x^6+x^3-x^8-x^5-x^2+x^5+x^2-x^4-x+1\right)\\ =\left(x^2+x+1\right)\left(x^9-x^8+x^6-x^4+x^3-x+1\right)\) 

\(e.x^8+x+1\\ =x^8-x^2+x^2+x+1\\ =x^2\left(x^6-1\right)+\left(x^2+x+1\right)\\ =x^2\left(x^3-1\right)\left(x^3+1\right)+\left(x^2+x+1\right)\\ =x^2\left(x-1\right)\left(x^2+x+1\right)\left(x^3+1\right)+\left(x^2+x+1\right)\\ =\left(x^2+x+1\right)\left[x^2\left(x-1\right)\left(x^3+1\right)+1\right]\\ =\left(x^2+x+1\right)\left[\left(x^3-x^2\right)\left(x^3+1\right)+1\right]\\ =\left(x^2+x+1\right)\left(x^6+x^3-x^5-x^2+1\right)\)

Bài 1:

a) $x^2+6x+9$

$=x^2+2.x.3+3^2$

$=(x+3)^2$

b) $9x^2-6x+1$

$=(3x)^2-2.3x.1+1^2$

$=(3x-1)^2$

c) $x^2y^2+xy+\frac14$

$=(xy)^2+2.xy.\frac12+\left(\frac12\right)^2$

$=\left(xy+\frac12\right)^2$

d) $(x-y)^2+6(x-y)+9$

$=(x-y)^2+2.(x-y).3+3^2$

$=(x-y+3)^2$

Bài 2: 

a) $-x^3+3x^2-3x+1$

$=1^3-3.1^2.x+3.1.x^2-x^3$

$=(1-x)^3$

b) $x^3+x^2+\frac13 x+\frac{1}{27}$

$=x^3+3.x^2.\frac13+3.x.\left(\frac13\right)^2+\left(\frac13\right)^3$

$=\left(x+\frac13\right)^3$

c) $x^6-3x^4y+3x^2y^2-y^3$

$=(x^2)^3-3.(x^2)^2.y+3.x^2.y^2-y^3$

$=(x^2-y)^3$

d) $(x-y)^3+(x-y)^2+\frac13 (x-y)+\frac{1}{27}$

$=(x-y)^3+3.(x-y)^2.\frac13+3.(x-y).\left(\frac13\right)^2+\left(\frac13\right)^3$

$=\left(x-y+\frac13\right)^3$

Bài 3:

a) $x^3+27$

$=x^3+3^3$

$=(x+3)(x^2-x.3+3^2)$

$=(x+3)(x^2-3x+9)$

b) $x^3-\frac18$

$=x^3-\left(\frac12\right)^3$

$=\left(x-\frac12\right)\left[x^2-x.\frac12+\left(\frac12\right)^2\right]$

$=\left(x-\frac12\right)\left(x^2-\frac12 x+\frac14\right)$

c) $8x^3+y^3$

$=(2x)^3+y^3$

$=(2x+y)[(2x)^2-2x.y+y^2]$

$=(2x+y)(4x^2-2xy+y^2)$

d) $8x^3-27y^3$

$=(2x)^3-(3y)^3$

$=(2x-3y)[(2x)^2+2x.3y+(3y)^2]$

$=(2x-3y)(4x^2+6xy+9y^2)$

Bài 4:

a) \(101^2=\left(100+1\right)^2\)

\(=100^2+2.100.1+1^2\)

\(=10000+200+1=10201\)

b) \(75^2-50.75+25^2\)

\(=75^2-2.75.25+25^2\)

\(=\left(75-25\right)^2\)

\(=50^2=2500\)

c) \(103.97\)

\(=\left(100+3\right).\left(100-3\right)\)

\(=100^2-3^2\\ =10000-9=9991\)

Bài 5:

a) \(\left(x+3y\right)^2-\left(x-3y\right)^2\)

\(=\left(x+3y-x+3y\right)\left(x+3y+x-3y\right)\\ =6y.2x=12xy\)

b) \(Q=\left(x-y\right)^2-4\left(x-y\right)\left(x+2y\right)+4\left(x+2y\right)^2\)

\(=\left(x-y\right)^2-2.\left(x-y\right).2\left(x+2y\right)+\left[2\left(x+2y\right)\right]^2\\ =\left[\left(x-y\right)-2\left(x+2y\right)\right]^2\\ =\left(x-y-2x-4y\right)^2\\ =\left(-x-5y\right)^2\)

c) \(A=\left(x+2\right)^3+\left(x-2\right)^3-2x\left(x^2+12\right)\)

\(=\left(x+2+x-2\right)\left[\left(x+2\right)^2-\left(x+2\right)\left(x-2\right)+\left(x-2\right)^2\right]-2x\left(x^2+12\right)\\ =2x\left(x^2+4x+4-x^2+4+x^2-4x+4\right)-2x\left(x^2+12\right)\\ =2x\left(x^2+12\right)-2x\left(x^2+12\right)=0\)

d) \(B=\left(xy+2\right)^3-6\left(xy+2\right)^2+12\left(xy+2\right)-8\)

\(=\left(xy+2\right)^3-3.\left(xy+2\right)^2.2+3.\left(xy+2\right).2^2-2^3\\ =\left(xy+2-2\right)^3\\ =\left(xy\right)^3=x^3y^3\)

e) \(A=\left(x-3\right)\left(x^2+3x+9\right)-\left(x^3+3\right)\)

\(=\left(x-3\right)\left(x^2+x.3+3^2\right)-x^3-3\\ =x^3-3^3-x^3-3\\ =-27-3=-30\)

Bài 6:

\(a,VT=\left(a-b\right)^2=a^2-2ab+b^2\\ =\left(a^2+2ab+b^2\right)-4ab\\ =\left(a+b\right)^2-4ab=VP\\ b,VT=\left(x+y\right)^2+\left(x-y\right)^2\\ =x^2+2xy+y^2+x^2-2xy+y^2\\ =2x^2+2y^2\\ =2\left(x^2+y^2\right)=VP\\ c,VT=\left(x+y\right)^2-\left(x-y\right)^2\\ =\left[\left(x+y\right)-\left(x-y\right)\right]\left[\left(x+y\right)+\left(x-y\right)\right]\\ =\left(x+y-x+y\right)\left(x+y+x-y\right)\\ =2y.2x=4xy=VP\\ d,VT=\left(x-y\right)^2+\left(x+y\right)^2+2\left(x^2-y^2\right)\\ =\left(x-y\right)^2+2\left(x-y\right)\left(x+y\right)+\left(x+y\right)^2\\ =\left[\left(x-y\right)+\left(x+y\right)\right]^2\\ =\left(x-y+x+y\right)^2\\ =\left(2x\right)^2=4x^2=VP\)

Bài 14:

1: \(A=x^2-x+3\)

\(=x^2-x+\dfrac{1}{4}+\dfrac{11}{4}\)

\(=\left(x-\dfrac{1}{2}\right)^2+\dfrac{11}{4}>=\dfrac{11}{4}\forall x\)

Dấu '=' xảy ra khi x-1/2=0

=>\(x=\dfrac{1}{2}\)

2: \(B=x^2+x+1\)

\(=x^2+x+\dfrac{1}{4}+\dfrac{3}{4}\)

\(=\left(x+\dfrac{1}{2}\right)^2+\dfrac{3}{4}>=\dfrac{3}{4}\forall x\)

Dấu '=' xảy ra khi \(x+\dfrac{1}{2}=0\)

=>\(x=-\dfrac{1}{2}\)

3: \(C=x^2-4x+1\)

\(=x^2-4x+4-3\)

\(=\left(x-2\right)^2-3>=-3\forall x\)

Dấu '=' xảy ra khi x-2=0

=>x=2

4: \(D=x^2-5x+7\)

\(=x^2-2\cdot x\cdot\dfrac{5}{2}+\dfrac{25}{4}+\dfrac{3}{4}\)

\(=\left(x-\dfrac{5}{2}\right)^2+\dfrac{3}{4}>=\dfrac{3}{4}\forall x\)

Dấu '=' xảy ra khi \(x-\dfrac{5}{2}=0\)

=>\(x=\dfrac{5}{2}\)

5: \(E=x^2+2x+2\)

\(=x^2+2x+1+1=\left(x+1\right)^2+1>=1\forall x\)

Dấu '=' xảy ra khi x+1=0

=>x=-1

6: \(F=x^2-3x+1\)
\(=x^2-2\cdot x\cdot\dfrac{3}{2}+\dfrac{9}{4}-\dfrac{5}{4}\)

\(=\left(x-\dfrac{3}{2}\right)^2-\dfrac{5}{4}>=-\dfrac{5}{4}\forall x\)

Dấu '=' xảy ra khi \(x-\dfrac{3}{2}=0\)

=>\(x=\dfrac{3}{2}\)

7: \(G=x^2+3x+3\)

\(=x^2+2\cdot x\cdot\dfrac{3}{2}+\dfrac{9}{4}+\dfrac{3}{4}\)

\(=\left(x+\dfrac{3}{2}\right)^2+\dfrac{3}{4}>=\dfrac{3}{4}\forall x\)

Dấu '=' xảy ra khi x+3/2=0

=>x=-3/2

8: \(H=3x^2+3-5x\)

\(=3\left(x^2-\dfrac{5}{3}x+1\right)\)

\(=3\left(x^2-2\cdot x\cdot\dfrac{5}{6}+\dfrac{25}{36}+\dfrac{11}{36}\right)\)

\(=3\left(x-\dfrac{5}{6}\right)^2+\dfrac{11}{12}>=\dfrac{11}{12}\forall x\)

Dấu '=' xảy ra khi x-5/6=0

=>x=5/6

9: \(I=4x+2x^2+3\)

\(=2\left(x^2+2x+\dfrac{3}{2}\right)\)

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

\(=2\left(x+1\right)^2+1>=1\forall x\)

Dấu '=' xảy ra khi x+1=0

=>x=-1

10: \(K=4x^2+3x+2\)

\(=\left(2x\right)^2+2\cdot2x\cdot\dfrac{3}{4}+\dfrac{9}{16}+\dfrac{23}{16}\)

\(=\left(2x+\dfrac{3}{4}\right)^2+\dfrac{23}{16}>=\dfrac{23}{16}\forall x\)

Dấu '=' xảy ra khi 2x+3/4=0

=>x=-3/8

11: M=(x-1)(x-3)+11

\(=x^2-4x+3+11=x^2-4x+14\)

\(=x^2-4x+4+10=\left(x-2\right)^2+10>=10\forall x\)

Dấu '=' xảy ra khi x-2=0

=>x=2

12: \(N=\left(x-3\right)^2+\left(x-2\right)^2\)

\(=x^2-6x+9+x^2-4x+4\)

\(=2x^2-10x+13\)

\(=2\left(x^2-5x+\dfrac{13}{2}\right)=2\left(x^2-5x+\dfrac{25}{4}+\dfrac{1}{4}\right)\)

\(=2\left(x-\dfrac{5}{2}\right)^2+\dfrac{1}{2}>=\dfrac{1}{2}\forall x\)

Dấu '=' xảy ra khi x-5/2=0

=>x=5/2

Bài 5:

Áp dụng BĐT Cô si cho 2 số: \(\sqrt{\dfrac{b+c}{a}}\) và 1

Có:

\(\sqrt{\dfrac{b+c}{a}}.1\le\dfrac{\left(\dfrac{b+c}{a}+1\right)}{2}\)

\(\Rightarrow\sqrt{\dfrac{b+c}{a}}\le\dfrac{a+b+c}{2a}\)

\(\Rightarrow\sqrt{\dfrac{a}{b+c}}\ge\dfrac{2a}{a+b+c}\)

Tương tự: \(\sqrt{\dfrac{b}{a+c}}\ge\dfrac{2b}{a+b+c}\)

\(\sqrt{\dfrac{c}{a+b}}\ge\dfrac{2c}{a+b+c}\)

\(\Rightarrow\sqrt{\dfrac{a}{b+c}}+\sqrt{\dfrac{b}{a+c}}+\sqrt{\dfrac{c}{a+b}}\ge\dfrac{2a}{a+b+c}+\dfrac{2b}{a+b+c}+\dfrac{2c}{a+b+c}\Rightarrow VT\ge2\Rightarrow VT>1\)