chứng minh : (ab + bd)2 + (ad - bc)2 = (a2 + b2).(c2 + d2)
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\(1,\left(ac+bd\right)^2+\left(ad-bc\right)^2\\ =a^2c^2+2abcd+b^2d^2+a^2d^2-2abcd+b^2c^2\\ =a^2c^2+b^2d^2+a^2d^2+b^2c^2\\ =\left(a^2c^2+a^2d^2\right)+\left(b^2d^2+b^2c^2\right)\\ =a^2\left(c^2+d^2\right)+b^2\left(c^2+d^2\right)\\ =\left(a^2+b^2\right)\left(c^2+d^2\right)\)
2, \(\left(a^2+b^2\right)\left(c^2+d^2\right)\ge\left(ac+bd\right)^2\)
\(\Leftrightarrow a^2c^2+b^2c^2+a^2d^2+b^2d^2\ge a^2c^2+2abcd+b^2d^2\)
\(\Leftrightarrow b^2c^2-2abcd+a^2d^2\ge0\)
\(\Leftrightarrow\left(bc-ad\right)^2\ge0\)
Dấu "=" xảy ra \(\Leftrightarrow bc=ad\Leftrightarrow\dfrac{a}{b}=\dfrac{c}{d}\)
\(1\)/
⇔ \(\left(ac\right)^2+2abcd+\left(bd\right)^2+\left(ad\right)^2-2abcd+\left(bc\right)^2=\left(a^2+b^2\right)\left(c^2+d^2\right)\)
⇔\(a^2\left(c^2+d^2\right)+b^2\left(c^2+d^2\right)=\left(a^2+b^2\right)\left(c^2+d^2\right)\)
⇔\(\left(a^2+b^2\right)\left(c^2+d^2\right)=\left(a^2+b^2\right)\left(c^2+d^2\right)\) ⇒ \(\left(dpcm\right)\)
\(2\)/
⇔\(\left(ac\right)^2+\left(ad\right)^2+\left(bc\right)^2+\left(bd\right)^2\ge\left(ac\right)^2+2abcd+\left(bd\right)^2\)
⇔\(\left(ad\right)^2-2abcd+\left(bc\right)^2\ge0\)
⇔\(\left(ad-bc\right)^2\ge0\left(đúng\right)\)
![](https://rs.olm.vn/images/avt/0.png?1311)
a: \(VT=a^2c^2+2abcd+b^2d^2+a^2d^2-2abcd+b^2c^2\)
\(=a^2c^2+a^2d^2+b^2d^2+b^2c^2\)
\(=a^2\left(c^2+d^2\right)+b^2\left(c^2+d^2\right)\)
\(=\left(c^2+d^2\right)\left(a^2+b^2\right)\)
b: Bạn ghi lại đề đi bạn
![](https://rs.olm.vn/images/avt/0.png?1311)
a: \(\left(ac+bd\right)^2+\left(ad-bc\right)^2\)
\(=a^2c^2+b^2d^2-2abcd+a^2d^2-2abcd+b^2c^2\)
\(=a^2c^2+a^2d^2+b^2d^2+b^2c^2\)
\(=\left(c^2+d^2\right)\left(a^2+b^2\right)\)
b: \(\left(ac+bd\right)^2< =\left(a^2+b^2\right)\left(c^2+d^2\right)\)
\(\Leftrightarrow a^2c^2+2abcd+b^2d^2-a^2c^2-a^2d^2-b^2c^2-b^2d^2< =0\)
\(\Leftrightarrow-a^2d^2+2abcd-b^2c^2< =0\)
\(\Leftrightarrow\left(ad-bc\right)^2>=0\)(luôn đúng)
a) \(\left(ac+bd\right)^2+\left(ad-bc\right)^2\)
\(=a^2c^2+2abcd+b^2d^2+a^2d^2-2adbc+b^2c^2\)
\(=a^2c^2+b^2d^2+a^2d^2+b^2c^2\)
\(=\left(a^2c^2+a^2d^2\right)+\left(b^2d^2+b^2c^2\right)\)
\(=a^2\left(c^2+d^2\right)+b^2\left(c^2+d^2\right)\)
\(=\left(a^2+b^2\right)\left(c^2+d^2\right)\)
b) \(\left(a^2+b^2\right)\left(c^2+d^2\right)-\left(ac+bd\right)^{^2}\)
\(=a^2c^2+a^2d^2+b^2c^2+b^2d^2-a^2c^2-2abcd-b^2d^2\)
\(=a^2d^2+b^2c^2-2abcd\)
\(=\left(ad\right)^2-2ad.bc+\left(bc\right)^2\)
\(=\left(ad-bc\right)^2\ge0\)
\(=\left(ac+bd\right)^2\le\left(a^2+b^2\right)\left(c^2+d^2\right)\)
![](https://rs.olm.vn/images/avt/0.png?1311)
Bài 1:
Biến đổi tương đương thôi:
\((ac+bd)^2+(ad-bc)^2=a^2c^2+b^2d^2+2abcd+a^2d^2+b^2c^2-2abcd\)
\(=a^2c^2+b^2d^2+a^2d^2+b^2c^2=(a^2+b^2)(c^2+d^2)\)
Ta có đpcm
Bài 2: Áp dụng kết quả bài 1:
\((a^2+b^2)(c^2+d^2)=(ac+bd)^2+(ad-bc)^2\geq (ac+bd)^2\) do \((ad-bc)^2\geq 0\)
Dấu bằng xảy ra khi \(ad=bc\Leftrightarrow \frac{a}{c}=\frac{b}{d}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(\left(ac+bd\right)^2+\left(ad-bc\right)^2=\left(a^2+b^2\right)\left(c^2+d^2\right)\left(1\right)\)
\(VT=a^2c^2+2abcd+b^2d^2+a^2d^2-2adbc+b^2c^2=a^2c^2+b^2d^2+a^2d^2+b^2c^2\)
\(VP=a^2c^2+a^2d^2+b^2c^2+b^2d^2\)
Ta thấy: \(VT=VP\)
\(\Rightarrow\left(1\right)\) luôn đúng.
Đề sai , xin sửa.
\(\left(ac+bd\right)^2+\left(ad-bc\right)^2\)
\(=a^2c^2+2abcd+b^2d^2+a^2d^2-2abcd+b^2c^2\)
\(=a^2c^2+b^2d^2+a^2d^2+b^2c^2\)
\(=a^2\left(c^2+d^2\right)+b^2\left(c^2+d^2\right)=\left(a^2+b^2\right)\left(c^2+d^2\right)\)
Xét vế trái : ( ab + cd )2 + ( ad - bc )2
= a2b2 + 2abcd + c2d2 + a2d2 - 2abcd + b2c2
= ( 2abcd - 2abcd ) + ( a2b2 + b2c2 ) + ( c2d2 + a2d2 )
= b2( a2 + c2 ) + d2( a2 + c2 )
= ( a2 + c2 )( b2 + d2 ) ( Vế phải )
Do đó : ( ab + cd )2 + ( ad + bc )2 = ( a2 + c2 )( b2 + d2 )