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

\(=x^2+x^2+1+3x^2+4+4x^2+9\)

\(=x^2+x^2+1+3x^2+3+4x^2+9+1\)

\(=2x^2+1+3x^2+3+4x^2+9+1\)

Từ đây ghép x vào rồi tính nốt đẳng thức thôi nhé

\(1,\\ a,=\left(x+2\right)\left(x^2-2x+4\right)\\ b,=\left(x-4\right)\left(x^2+8x+16\right)\\ c,=\left(3x+1\right)\left(9x^2-3x+1\right)\\ d,=\left(4m-3\right)\left(16m^2+12m+9\right)\\ 2,\\ a,=x^3+125\\ b,=1-x^3\\ c,=y^3+27t^3\)

a)
\(=\left(x+2\right)\left(x^2-2x+4\right)\)
b)
\(=\left(x-4\right)\left(x^2+4x+16\right)\)
c)=\(\left(3x+1\right)\left(9x^2-3x+1\right)\)
d)
=\(\left(4m-3\right)\left(16m^2+12m+9\right)\)

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

\(=x^2+2\left(x^2+2x+1\right)+3\left(x^2-4x+4\right)+4\left(x^2+6x+9\right)\)

\(=x^2+2x^2+4x+2+3x^2-12x+12+4x^2+24x+36\)

\(=10x^2+16x+50\)

( a² - 2a + 3 )( a² + 2a - 3 )

= [ a² - ( 2a - 3 ) ][ a² + ( 2a - 3 ) ]

= ( a² )² - ( 2a - 3 )²

= a⁴ - ( 4a² -  12a + 9 )

= a⁴ - 4a² + 12a - 9 

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

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

\(=a^4-4a^2+12a-9\)

( 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

1.a) xy + 2y - x² + 4

= y ( x + 2 ) - ( x² - 4 ) = y ( x + 2 ) - ( x - 2 ) ( x + 2 ) = ( x + 2 )( y - x + 2 )

b) 2x² + y² + 3xy

= ( 2x² + 2xy ) + ( y² + xy )

= 2x ( x + y ) + y ( x + y )

= ( x + y ) ( 2x + y )

2.

x - y = 5 \(\Rightarrow\)( x - y )² = 25 \(\Rightarrow\)x² + y² = 25 + 2xy = 25 + 2.3 = 31

A = ( x + y )² = x² + y² + 2xy = 31 + 6 = 37

Câu 7:

a: \(A=x^2-2x+7\)

\(=x^2-2x+1+6\)

\(=\left(x-1\right)^2+6\ge6\forall x\)

Dấu '=' xảy ra khi x-1=0

=>x=1

b: \(B=5x^2-20x\)

\(=5\left(x^2-4x\right)\)

\(=5\left(x^2-4x+4-4\right)=5\left(x-2\right)^2-20\ge-20\forall x\)

Dấu '=' xảy ra khi x-2=0

=>x=2

Câu 4:

a: \(A=\left(x-y\right)^2+\left(x+y\right)^2\)

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

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

b: \(B=\left(2x-1\right)^2-2\left(2x-3\right)^2+4\)

\(=4x^2-4x+1-2\left(4x^2-12x+9\right)+4\)

\(=4x^2-4x+5-8x^2+24x-18=-4x^2+20x-13\)

Câu 2:

a: \(x^2-4x+4=x^2-2\cdot x\cdot2+2^2=\left(x-2\right)^2\)

b: \(x^2+10x+25=x^2+2\cdot x\cdot5+5^2=\left(x+5\right)^2\)

c: \(\frac{x^2}{4}-x+1=\left(\frac12x\right)^2-2\cdot\frac12x\cdot1+1^2=\left(\frac12x-1\right)^2\)

d: \(9\left(x+1\right)^2-6\left(x+1\right)+1\)

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

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

e: \(\left(x-2y\right)^2-8\left(x^2-2xy\right)+16x^2\)

\(=\left(x-2y\right)^2-2\cdot\left(x-2y\right)\cdot4x+\left(4x\right)^2\)

\(=\left(x-2y-4x\right)^2=\left(-3x-2y\right)^2\)