( 1/2 + x )² = ( 1/2 )² + 2.1/2.x + x² = x² + x + 1/4

( 2x + 1 )² = ( 2x )² + 2.2x.1 + 1² = 4x² + 4x + 1

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

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

\(=x^4+4x^3-2x^2+4x^2+4x+1\)

\(=x^4+4x^3+2x^2+4x+1\)

a.

\(\frac{x^2}{4}+x+3=\frac{x^2}{4}+x+1+2=\left(\frac{x}{2}+1\right)^2+2>0;\forall x\)

b.

\(A=-3x^2+2x-5=-3\left(x^2-2.\frac{1}{3}x+\frac{1}{9}\right)-\frac{14}{3}=-3\left(x-\frac{1}{3}\right)^2-\frac{14}{3}\le-\frac{14}{3}\)

\(A_{max}=-\frac{14}{3}\) khi \(x=\frac{1}{3}\)

c.

Đề thiếu (để ý 2 số hạng cuối)

\(A=x^4-2x^3+x^2+3x^2-6x+3-1\)

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

\(A_{min}=-1\) khi \(x=1\)

d.

\(27x^2-\frac{9}{2}x+\frac{3}{16}=3\left(9x^2-\frac{3}{2}x+\frac{1}{16}\right)=3\left(3x-\frac{1}{4}\right)^2\)

e.

\(=\left[\left(b+c\right)+a\right]^2+\left[\left(b+c\right)-a\right]^2+\left[a-\left(b-c\right)\right]^2+\left[a+\left(b-c\right)\right]^2\)

\(=2\left(b+c\right)^2+2a^2+2a^2+2\left(b-c\right)^2\)

\(=4a^2+2b^2+4bc+2c^2+2b^2-4bc+2c^2\)

\(=4\left(a^2+b^2+c^2\right)\)

f.

\(\left(a^2+b^2\right)\left(c^2+d^2\right)=a^2c^2+b^2d^2+a^2d^2+b^2c^2\)

\(=\left(a^2c^2+b^2d^2+2ac.bd\right)+\left(a^2d^2+b^2c^2-2ad.bc\right)\)

\(=\left(ac+bd\right)^2+\left(ad-bc\right)^2\)

a) \(\left(x^2-2x-1\right)^2\)

\(=\left[x^2+\left(-2x\right)+\left(-1\right)\right]\left[x^2+\left(-2\right)+\left(-1\right)\right]\)

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

\(=x^4-2x^3-x^2-2x^3+4x^3+2x-x^2+2x+1\)

\(=x^4-4x^3+2x^2+4x+1\)

Mk ko chắc

a) \(\left(x^2-2x-1\right)^2\)

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

\(=x^4+4x^3-2x^2+4x^2+4x+1\)

\(=x^4+4x^3-2x^2+4x+1\)

b) Tương tự

2:

-8x^6-12x^4y-6x^2y^2-y^3

=-(8x^6+12x^4y+6x^2y^2+y^3)

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

3:

=(1/3)^2-(2x-y)^2

=(1/3-2x+y)(1/3+2x-y)

`a,-x^3/8 + 3/(4x^2) - 3/(2x) +1`

`=-(x^3/8 - 3/(4x^2) + 3/(2x) - 1)`

`=-(x/2 - 1)^3`

`b,x^6 - 3/(2x^{4} y) + 3/(4x^{2}y^{2}) - 1/(8y^{3})`

`=(x^3 - 1/(2y))^{3}`

MINH CUNG VAY

a) Ta có: \(\left(x^2+9x+18\right)^2+2\left(x^2+9x\right)+37\)

\(=\left(x^2+9x+18\right)^2+2\cdot\left(x^2+9x+18\right)-36+37\)

\(=\left(x^2+9x+19\right)^2\)

b) Ta có: \(x^2+y^2+2x+2y+2\left(x+1\right)\left(y+1\right)+2\)

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

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

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

\(=x^2+2\cdot x\cdot\left(y+2\right)+\left(y+2\right)^2\)

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

d) Ta có: \(x^2+2x\left(y+1\right)+y^2+2y+1\)

\(=x^2+2\cdot x\cdot\left(y+1\right)+\left(y+1\right)^2\)

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