Bài 3: Những hằng đẳng thức đáng nhớ

\(x^2-6=x^2-\left(\sqrt{6}\right)^2=\left(x-\sqrt{6}\right)\left(x+\sqrt{6}\right)\)

a² + b² + c² + d² + e² \(\ge\) a(b+c+d+e)

Xét: 4(a²+b²+c²+d²+e²) - 4(ab+ac+ad+ae)

= 4a² + 4b² + 4c² + 4d² + 4e² - 4ab - 4ac - 4ad - 4ae

= (a²+4b²-4ab) + (a²+4c²-4ac) + (a²+4d²-4ad) + (a²+4e²-4ae)

= (a-2b)² + (a-2c)² + (a-2d)² + (a-2e)² \(\ge\) 0

=> 4(a²+b²+c²+d²+e²) \(\ge\) 4(ab+ac+ad+ae)

=> a²+b²+c²+d²+e² \(\ge\) ab + ac + ad + ae

Có: x² + y² + z² = xy + yz + zx

<=> 2x² + 2y² + 2z² - 2xy - 2yz - 2zx = 0

<=> (x² + y² - 2xy) + (y² + z² - 2yz) + (z² + x² - 2zx) = 0

<=> (x-y)² + (y-z)² + (z-x)² = 0

<=> x-y = y-z = z-x = 0

<=> x = y = z

\(x^2+y^2+z^2=xy+yz+xz\\ 2x^2+2y^2+2z^2=2xy+2yz+2xz\\ 2x^2+2y^2+2z^2-2xy-2yz-2xz=0\\ x^2-2xy+y^2+y^2-2yz+z^2+z^2+-2zx+x^2=0\\ \left(x-y\right)^2+\left(y-z\right)^2+\left(z-x\right)^2=0\\ \Rightarrow\left\{{}\begin{matrix}x-y=0\\y-z=0\\z-x=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x=y\\y=z\\z=x\end{matrix}\right.\Rightarrow x=y=z\)

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

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

\(=4\)

\(\Rightarrow\)Giá trị của biểu thức ko phụ thuộc vào biến

\(\Rightarrowđpcm\)

Bài 2 :

\(4x^2+4xy+y^2=\left(2x+y\right)^2\)

\(9m^2+n^2-6mn=\left(3m-n\right)^2\)

\(16a^2+25b^2+40ab=\left(4a+5b\right)^2\)

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

Bài 1:

a, ( 3 + xy²)² = 3² + 2. 3. xy² + (xy²)² = 9+ 6xy² + x²y⁴.

b, (10- 2m²ⁿ) = 10² - 2.10.2m²ⁿ + (2m²ⁿ)² = 100 - 40m²ⁿ + 4m⁴ⁿ

c, ( a - b²)(a+b²) = bạn xem lại đề câu này nhé!

Bài 2:

a, 4x² + 4xy + y² = (2x)² + 2. 2x. y + y² = ( 2x + y)²

b, 9m² + n² - 6mn = ( 3m)² - 2. 3m. n + n² = ( 3m - n)²

c,16a² + 25b² +40ab = (4a)² + 2. 4a. 5b + (5b)² = ( 4a + 5b) ²

d, x² - x +1/4 = x² - 2. 1/2. x +(1/2)² = (x - 1/2)²

Bạn hok đồng xuân à

a,\(401^2=\left(400+1\right)^2=400^2+2\cdot400+1\)

\(=160000+800+1=160801\)

b, \(999^2=\left(1000-1\right)^2=1000^2-2\cdot1000+1\)

\(=1000000-2000+1=998001\)

c, \(77\cdot83=\left(80-3\right)\left(80+3\right)=80^2-3^2\)

\(=6400-9=6391\)

(2a+1)² = 4a² + 4a + 1

(2x-y)² = 4x² - 4xy + y²

(x-2y)(x+2y) = x² - 4y²