Bài 3: Những hằng đẳng thức đáng nhớ

x³ - 27 = x³ - 3³

= (x - 3)(x² + 3x + 3²)

= (x - 3)(x² + 3x + 9)

(x + y)² = x² + 2xy + y²

\(\left(A+B\right)^2=A^2+2\cdot A\cdot B+B^2\)

\(\left(A+B\right)^2=A^2+2AB+B^2\)

Lời giải:

$x^2+4x-8y-2xy+2y^2+2024$

$=(x^2-2xy+y^2)+4x-8y+y^2+2024$

$=(x-y)^2+4(x-y)+y^2-4y+2024$

$=(x-y)^2+4(x-y)+4+(y^2-4y+4)+2016$

$=(x-y+2)^2+(y-2)^2+2016\geq 2016$

Vậy GTNN của biểu thức là 2016. Giá trị này đạt tại $x-y+2=y-2=0$

$\Leftrightarrow y=2; x=0$

a: \(A=2x^2-5x-2x\left(x+1\right)\)

\(=2x^2-5x-2x^2-2x\)

=-7x

b: \(B=\left(x+2y\right)\left(x-2y\right)-4y\left(x-y\right)\)

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

\(=x^2-4xy\)

c: \(C=\left(2x-8\right)\left(x^2+4x+16\right)-2x\left(x^2-2\right)\)

\(=2\left(x-4\right)\left(x^2+4x+16\right)-2x\left(x^2-2\right)\)

\(=2\left(x^3-64\right)-2x^3+4x\)

\(=2x^3-128-2x^3+4x=4x-128\)

a) \((a+y)^3=a^3+3a^2y+3ay^2+y^3\)

b) \((x-b)^3=x^3-3x^2b+3xb^2-b^3\)

a: \(\left(2n+1\right)^2-\left(2n-1\right)^2\)

\(=\left(2n+1-2n+1\right)\left(2n+1+2n-1\right)\)

\(=2\cdot4n=8n⋮8\)

b: \(\left(8n+4\right)^2-\left(2n+1\right)^2\)

\(=16\left(2n+1\right)^2-\left(2n+1\right)^2\)

\(=15\left(2n+1\right)^2⋮15\)

a: \(\left(a+b\right)^2-\left(a-b\right)^2\)

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

\(=a^2+2ab+b^2-a^2+2ab-b^2\)

=4ab

b: \(\left(a+b\right)\left[\left(a-b\right)^2+ab\right]\)

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

\(=\left(a+b\right)\left(a^2-ab+b^2\right)=a^3+b^3\)

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

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

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

a)

\(P=(x-10)^2-x(x+80)\\=x^2-2\cdot x\cdot10+10^2-x^2-80x\\=x^2-20x+100-x^2-80x\\=-100x+100\\=-100(x-1)\)

Thay \(x=0,87\) vào \(P\), ta được:

\(P=-100(0,87-1)=-100\cdot(-0,13)=13\)

\(---\)

b)

\(Q=4a^2+8ab+4b^2\\=4(a^2+2ab+b^2)\\=4(a+b)^2\)

Thay \(a=65;b=35\) vào \(Q\), ta được:

\(Q=4(65+35)^2\\=4\cdot 100^2\\=4\cdot 10000\\=40000\)

\(---\)

c)

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

Thay \(x=101\) vào \(R\), ta được:

\(R=(101-1)^3\\=100^3\\=1000000\)

\(Toru\)

a: \(P=\left(x-10\right)^2-x\left(x+80\right)\)

\(=x^2-20x+100-x^2-80x=-100x+100\)

Khi x=0,87 thì \(P=100-100\cdot0.87=100-87=13\)

c: \(R=x^3-3x^2+3x-1\)

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

Khi x=101 thì \(R=\left(101-1\right)^3=100^3=1000000\)

b: \(Q=4a^2+8ab+4b^2\)

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

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

Khi a=65 và b=35 thì \(Q=4\left(35+65\right)^2=4\cdot100^2=40000\)

a)

\(20x^2-(5x-4)(4+5x)\\=20x^2-(5x-4)(5x+4)\\=20x^2-[(5x)^2-4^2]\\=20x^2-(25x^2-16)\\=20x^2-25x^2+16\\=-5x^2+16\)

b)

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

c)

\((x+3)^3(x-3)^3\\=[(x+3)(x-3)]^3\\=(x^2-3^2)^3\\=(x^2-9)^3\\=(x^2)^3-3\cdot (x^2)^2\cdot9+3\cdot x^2\cdot9^2-9^3\\=x^6-27x^4+243x^2-729\)

d)

\(x(x-1)(x+1)-(x-3)(x^2+3x+9)\\=x(x^2-1^2)-(x-3)(x^2+x\cdot3+3^2)\\=x^3-x-(x^3-3^3)\\=x^3-x-x^3+27\\=-x+27\\Toru\)

a: \(20x^2-\left(5x-4\right)\left(5x+4\right)\)

\(=20x^2-\left(25x^2-16\right)\)

\(=20x^2-25x^2+16=-5x^2+16\)

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

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

\(=y^2-4xy\)

c: \(\left(x+3\right)^3\cdot\left(x-3\right)^3\)

\(=\left[\left(x+3\right)\left(x-3\right)\right]^3\)

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

\(=x^6-27x^4+243x^2-729\)

d: \(x\left(x-1\right)\left(x+1\right)-\left(x-3\right)\left(x^2+3x+9\right)\)

\(=x\cdot\left(x^2-1\right)-\left(x^3-27\right)\)

\(=x^3-x-x^3+27=27-x\)