1)

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

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

2)

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

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

3)

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

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

4)

\(=x^2-2xy+y^2+x^2+10x+25\)

\(=\left(x-y\right)^2+\left(x+5\right)^2\)

5)

\(=a^2+2ab+b^2+4b^2+4b+1\)

\(=\left(a+b\right)^2+\left(2b+1\right)^2\)

1/ x² - 4x + 5 + y² + 2y 

= ( x² - 4x + 4 ) + ( y² + 2y + 1 )

= ( x - 2 )² + ( y + 1 )²

2/ 2a² + 2ab - 2ax + x² + b²

= ( a² + 2ab + b² ) + ( x² - 2ax + a² )

= ( a + b )² + ( x - a )²

3/ x² - 2x + y² + 6y + 10

= ( x² - 2x + 1 ) + ( y² + 6y + 9 )

= ( x - 1 )² + ( y + 3 )²

4/ 2x² + y² - 2xy + 10x + 25

= ( x² - 2xy + y² ) + ( x² + 10x + 25 )

= ( x - y )² + ( x + 5 )²

5/ a² + 2ab + 5b² + 4b + 1

= ( a² + 2ab + b² ) + ( 4b² + 4b + 1 )

= ( a + b )² + ( 2b + 1 )²

BÀI 1:

\(A=\left(x-10\right)^2+103\)

Có:    \(\left(x-10\right)^2\ge0\forall x\)

=>   \(A\ge103\)

DẤU "=" XẢY RA <=>   \(\left(x-10\right)^2=0\Rightarrow x=10\)

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

Có:   \(\left(2x+1\right)^2\ge0\forall x\)

=>   \(B\ge-6\)

DẤU "=" XẢY RA <=>   \(\left(2x+1\right)^2=0\Leftrightarrow x=-\frac{1}{2}\)

BÀI 3:

a) \(A=y^4+y^3-y^2-2y-\left(y^4+y^3+y^2-2y^2-2y-2\right)\)

\(A=y^4+y^3-y^2-2y-y^4-y^3+y^2+2y+2\)

\(A=2\)

b)   \(B=\left(2x\right)^3+3^3-8x^3+2\)

\(B=29\)

Bài 1.

A = x² - 20x + 103

A = ( x² - 20x + 100 ) + 3

A = ( x - 10 )² + 3 ≥ 3 ∀ x

Đẳng thức xảy ra <=> x - 10 = 0 => x = 10

=> MinA = 3 <=> x = 10

B = 4x² + 4x - 5

B = ( 4x² + 4x + 1 ) - 6

B = ( 2x + 1 )² - 6 ≥ -6 ∀ x

Đẳng thức xảy ra <=> 2x + 1 = 0 => x = -1/2

=> MinB = -6 <=> x = -1/2

Bài 2.

A = -x² + 8x - 21

A = -x² + 8x - 16 - 5

A = -( x² - 8x + 16 ) - 5

A = -( x - 4 )² - 5 ≤ -5 ∀ x

Đẳng thức xảy ra <=> x - 4 = 0 => x = 4

=> MaxA = -5 <=> x = 4

B = lỗi đề :>

Bài 3.

a) y( y³ + y² - y - 2 ) - ( y² - 2 )( y² + y + 1 )

= y⁴ + y³ - y² - 2y - ( y⁴ + y³ + y² - 2y² - 2y - 2 )

= y⁴ + y³ - y² - 2y - y⁴ - y³ - y² + 2y² + 2y + 2

= 2 ( đpcm )

b) ( 2x + 3 )( 4x² - 6x + 9 ) - 2( 4x³ - 1 )

= ( 2x )³ + 27 - 8x³ + 2

= 8x³ + 27 - 8x³ + 2

= 29 ( đpcm )

1. Ta có: \(\left(a+b\right)^2-\left(a-b\right)^2=\left(a+b+a-b\right)\left(a+b-a+b\right)\)

\(=2a.2b=4ab\)

=> đpcm

2. Ta có: \(\left(a+b\right)^2+\left(a-b\right)^2=a^2+2ab+b^2+a^2-2ab+b^2\)

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

=> đpcm

3. Ta có:\(\left(a+b\right)^2-4ab=a^2+2ab+b^2-4ab\)

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

=> đpcm

4. Ta có: \(\left(a-b\right)^2+4ab=a^2-2ab+b^2+4ab\)

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

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

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

\(\Leftrightarrow a^2+b^2-a^2-b^2+2ab+2ab=4ab\)

\(\Leftrightarrow4ab=4ab\Leftrightarrow4ab-4ab=0\Leftrightarrow0=0\)(đpcm)

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

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

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

\(\Leftrightarrow2\left(a^2+b^2\right)=2\left(a^2+b^2\right)\Leftrightarrow2\left(a^2+b^2\right)-2\left(a^2+b^2\right)=0\Leftrightarrow0=0\)(đpcm)

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

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

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

\(\Leftrightarrow\left(a-b\right)^2=\left(a-b\right)^2\Leftrightarrow\left(a-b\right)^2-\left(a-b\right)^2=0\Leftrightarrow0=0\)(đpcm)

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

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

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

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

\(\Leftrightarrow\left(a+b\right)^2-\left(a+b\right)^2=0\Leftrightarrow0=0\)(đpcm)

Đặt BT trên  là A

Ta có : 

\(A^2=5-\sqrt{21}+5+\sqrt{21}+2\sqrt{(5-\sqrt{21})\left(5+\sqrt{21}\right)}\)

\(=10+2\sqrt{25-21}\)

\(=10+2.\sqrt{4}=10+2.2=14\)

\(\Rightarrow A=\sqrt{14}\)

Ta có:

\(\sqrt{5-\sqrt{21}}+\sqrt{5+\sqrt{21}}\)

\(=\sqrt{\left(\sqrt{5-\sqrt{21}}+\sqrt{5+\sqrt{21}}\right)^2}\)

\(=\sqrt{5-\sqrt{21}+2\sqrt{\left(5-\sqrt{21}\right)\left(5+\sqrt{21}\right)}+5+\sqrt{21}}\)

\(=\sqrt{10+2\sqrt{25-21}}\)

\(=\sqrt{10+2\sqrt{4}}=\sqrt{10+4}=\sqrt{14}\)

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

\(=x^2+10x+25-x^2+10x-25-20x+2\)

\(=2\) không phụ thuộc vào \(x\)

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

\(=x^2-2x-15-x^2+2x-1\)

\(=-16\) không phụ thuộc vào \(x\)

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

\(=3x^2-4x-4-3x^2+5x+8\)

\(=x+8\) câu này đề sai.

d) \(2.\left(3x+1\right)\left(2x+5\right)-6x.\left(2x+4\right)-10\left(x-1\right)\)

\(=2.\left(6x^2+17x+5\right)-\left(12x^2+24x\right)-10x+10\)

\(=12x^2+34x+10-12x^2-24x-10x+10\)

\(=20\) không phụ thuộc vào \(x\)

a) ( x + 5 )² - ( x - 5 )² - 20x + 2 

= x² + 10x + 25 - ( x² - 10x + 25 ) - 20x + 2

= x² + 10x + 25 - x² + 10x - 25 - 20x + 2

= 2 ( đpcm )

b) ( x + 3 )( x - 5 ) - ( x - 1 )²

= x² - 2x - 15 - ( x² - 2x + 1 )

= x² - 2x - 15 - x² + 2x - 1

= -16 ( đpcm )

c) ( 3x + 2 )( x - 2 ) - x( 3x - 5 ) + 8

= 3x² - 4x - 4 - 3x² + 5x + 8

= x + 4 ( lỗi đề )

d) 2( 3x + 1 )( 2x + 5 ) - 6x( 2x + 4 ) - 10( x - 1 )

= 2( 6x² + 17x + 5 ) - 12x² - 24x - 10x + 10

= 12x² + 34x + 10 - 12x² - 24x - 10x + 10

= 20 ( đpcm )

\(2x\left(x-5\right)-x\left(3+2x\right)=26\)

\(\Leftrightarrow2x^2-10x-3x-2x^2=26\)

\(\Leftrightarrow-13x=26\)

\(\Leftrightarrow x=-2\)

Bài làm:

Ta có: \(2x\left(x-5\right)-x\left(3+2x\right)=26\)

\(\Leftrightarrow2x^2-10x-3x-2x^2-26=0\)

\(\Leftrightarrow-13x=26\)

\(\Rightarrow x=-2\)

Ta có : \(B=x.\left(x^2+x+1\right)-x^2.\left(x+1\right)-x+5\)

\(=x^3-x^2-x-x^3-x^2-x+5\)

\(=5\) không phụ thuộc vào giá trị biến \(x\)

B = x( x² + x + 1 ) - x²( x + 1 ) - x + 5

B = x³ + x² + x - x³ - x² - x + 5

B = 5 

=> đpcm

a.x = -3

b.\(\frac{13}{2}\)

c. (x+4)²

d. không bik

Bài làm:

a) \(\left(x+4\right)^2-1=0\)

\(\Leftrightarrow\left(x+4\right)^2=1\)

\(\Leftrightarrow\orbr{\begin{cases}x+4=1\\x+4=-1\end{cases}}\Rightarrow\orbr{\begin{cases}x=-3\\x=-5\end{cases}}\)

b) \(\left(2x-3\right)^2=100\)

\(\Leftrightarrow\orbr{\begin{cases}2x-3=10\\2x-3=-10\end{cases}}\Leftrightarrow\orbr{\begin{cases}2x=13\\2x=-7\end{cases}}\Rightarrow\orbr{\begin{cases}x=\frac{13}{2}\\x=-\frac{7}{2}\end{cases}}\)

c) \(x^2+8x+16=0\)

\(\Leftrightarrow\left(x+4\right)^2=0\)

\(\Rightarrow x+4=0\)

\(\Rightarrow x=-4\)

d) \(4x^2-12x=-9\)

\(\Leftrightarrow4x^2-12x+9=0\)

\(\Leftrightarrow\left(2x-3\right)^2=0\)

\(\Rightarrow2x-3=0\)

\(\Rightarrow x=\frac{3}{2}\)

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