Chứng Minh : (a^2 - b^2)( d^2-c^2)= ( ad+bc)^2 - ( ac+bd)^2
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\(VT=a^2c^2+2abcd+b^2d^2+a^2d^2-2abcd+b^2c^2\)
\(=a^2c^2+a^2d^2+b^2d^2+b^2c^2\)
\(=\left(c^2+d^2\right)\cdot\left(a^2+b^2\right)\)
\((ac + bd)^2 + (ad – bc)^2 = (ac)^2 +(bd)^2 + 2(ac)(bd) + (ad)^2 +(bc)^2 - 2(ad)(bc) \)
\( = (ac)^2 +(bd)^2 + (ad)^2 +(bc)^2 + 2abcd – 2abcd\)
\(= a^2c^2 + b^2d^2 + a^2d^2 + b^2c^2\)
\( = (a^2 + b^2)(c^2 + d^2)\)
➤ \((ac + bd)^2 + (ad – bc)^2 = (a^2 + b^2)(c^2 + d^2)\)
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(\left(a^2+b^2\right)\left(c^2+d^2\right)=a^2c^2+a^2d^2+b^2c^2+b^2d^2=\left(a^2c^2+2abcd+b^2d^2\right)+\left(a^2d^2-2abcd+b^2c^2\right)\)
\(=\left(ac+bd\right)^2+\left(ad-bc\right)^2\)
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(VT=\left(ac\right)^2+\left(bc\right)^2+\left(ad\right)^2+\left(bd\right)^2\)
\(VP=\left(ac^2\right)+2acbd+\left(bd\right)^2+\left(ad\right)^2-2adbc+\left(bc\right)^2=\left(ac\right)^2+\left(bc\right)^2+\left(ad\right)^2+\left(bd\right)^2\)
\(VT=VP\)
Ta sẽ biến đổi vế phải bằng vế trái :
Ta có :
\(\left(ac+bd\right)^2+\left(ad-bc\right)^2=\left(a^2c^2+b^2d^2+2abcd\right)+\left(a^2d^2+b^2c^2-2abcd\right)=\left(a^2c^2+b^2c^2\right)+\left(b^2d^2+a^2d^2\right)=c^2\left(a^2+b^2\right)+d^2\left(a^2+b^2\right)=\left(a^2+b^2\right)\left(c^2+d^2\right)\)
Vậy \(\left(a^2+b^2\right)\left(c^2+d^2\right)=\left(ac+bd\right)^2+\left(ad-bc\right)^2\)
![](https://rs.olm.vn/images/avt/0.png?1311)
a) Cách lầy lội nhất khai triển hết ra :|
\(\left(ac+bd\right)^2+\left(ad-bc\right)^2=a^2c^2+2abcd+b^2d^2+a^2d^2-2abcd+b^2c^2\)
\(=\left(a^2c^2+b^2c^2\right)+\left(b^2d^2+a^2d^2\right)=c^2\left(a^2+b^2\right)+d^2\left(a^2+b^2\right)=\left(a^2+b^2\right)\left(c^2+d^2\right)\)
a) \(\left(ac+bd\right)^2+\left(ad-bc\right)^2=\left(a^2+b^2\right)\left(c^2+d^2\right)\)
\(\Leftrightarrow\left(ac+bd\right)^2+\left(ad-bc\right)^2=a^2c^2+a^2d^2+b^2c^2+b^2d^2\)
Biến đổi vế traias ta có:
\(\left(ac+bd\right)^2+\left(ad-bc\right)^2=a^2c^2+2abcd+b^2d^2+a^2d^2-2abcd+b^2c^2\)
\(=a^2c^2+b^2d^2+a^2d^2+b^2c^2=VP\)
=>đpcm
b)Có: \(\left(ac+bd\right)^2\le\left(a^2+b^2\right)\left(c^2+d^2\right)\)
\(\Leftrightarrow a^2c^2+2abcd+b^2d^2\le a^2c^2+a^2d^2+b^2c^2+b^2d^2\)
\(\Leftrightarrow-a^2d^2+2abcd-b^2c^2\le0\)
\(\Leftrightarrow-\left(a^2d^2-2abcd+b^2c^2\right)\le0\)
\(\Leftrightarrow-\left(ad-bc\right)^2\le0\), luôn luôn đúng
=>đpcm
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
a/
Đẳng thức <=> (ac)² + (ad)² + (bc)² + (bd)² = (ac)² + (ad)² + (bc)² + (bd) + 2ac.bd - 2ad.bc
<=> 2.ad.bc - 2.ad.bc = 0
<=> 0 = 0 ( đúng ) => đẳng thức đã cho đúng
b/
Đẳng thức <=> 2a² + 2b² + 2c² = 2ab + 2bc + 2ac
<=> a² - 2ab + b² + b² - 2bc + c² + c² - 2ac + a² = 0
<=> ( a - b)² + ( b - c)² + ( c - a)² = 0
<=> (a - b)² = 0 và (b - c)² = 0 và (c - a)² = 0
<=> a - b = 0 và b - c = 0 và c - a = 0
<=> a = b, b = c, c = a => a = b = c
(vì tổng 3 số hk âm = 0 khi mỗi số điều = 0)
c/ từ giả thuyết => a + b = -c,
ta có:
a³ + b³ + c³ -3abc = ( a + b)³ - 3ab( a + b) + c³ -3abc = -c³ + 3abc + c³ - 3abc = 0
( vì a³ + b³ = ( a + b)( a² - ab + b²) = (a + b)( (a + b)² - 3ab ) = ( a + b)³ - 3ab( a + b)
=> ĐPCM
\(\left(a^2-b^2\right)\left(d^2-c^2\right)=\left(ad+bc\right)^2-\left(ac+bd\right)^2\)
\(\Leftrightarrow a^2d^2-a^2c^2-b^2d^2+b^2c^2=a^2d^2+2adbc+b^2c^2-a^2c^2-2acbd-b^2d^2\)
\(\Leftrightarrow a^2d^2-a^2c^2-b^2d^2+b^2c^2=a^2d^2+b^2c^2-a^2c^2-b^2d^2\)
\(\Leftrightarrow-a^2c^2-b^2d^2+b^2c^2=b^2c^2-a^2c^2-b^2d^2\)
\(\Leftrightarrow-b^2d^2+b^2c^2=b^2c^2-b^2d^2\)
\(\Leftrightarrow b^2c^2=b^2c^2\)