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- Bài 1.
a) \(\left(5x+1\right)^2-\left(5x-3\right)\left(5x+3\right)=30\)
\(\Leftrightarrow\left(25x^2+10x+1\right)-25x^2+9=30\)
\(\Leftrightarrow10x=20\Leftrightarrow x=2\)
b) \(\left(x-1\right)\left(x^2+x+1\right)-x\left(x+2\right)\left(x-2\right)=5\)
\(\Leftrightarrow x^3-1-x^3+4x-5=0\)
\(\Leftrightarrow4x=6\Leftrightarrow x=\frac{3}{2}\)
- Ta có : \(A=1997.1999=\left(1998-1\right)\left(1998+1\right)=1998^2-1< 1998^2\)
\(\Rightarrow A< B\)
- Từ a+b+c=2p => \(p=\frac{a+b+c}{2}\)
Ta có : \(4p\left(p-a\right)=2\left(a+b+c\right)\left(\frac{a+b+c}{2}-a\right)=2.\left(a+b+c\right).\frac{b+c-a}{2}\)
\(=\left(a+b+c\right)\left(b+c-a\right)=\left[\left(b+c\right)+a\right]\left[\left(b+c\right)-a\right]=\left(b+c\right)^2-a^2\)
\(=b^2+c^2-a^2+2bc\)
Bài cuối bạn sửa 2ab thành 2bc nhé ^^
\(2bc+b^2+c^2-a^2\)
\(=\left(b+c\right)^2-a^2\)
\(=\left(b+c+a\right)\cdot\left(b+c-a\right)\)
\(=2p\cdot\left(2p-a-a\right)\)
\(=4p\left(p-a\right)\)
Sửa đề : CMR : \(2bc+b^2+c^2-a^2=4p\left(p-a\right)\)
Ta có : \(VT=2bc+b^2+c^2-a^2\)
\(=\left(b+c\right)^2-a^2\)
\(=\left(b+c-a\right)\left(b+c+a\right)\)
\(=\left(b+c+a-2a\right).2p\)
\(=\left(2p-2a\right).2p\)
\(=4p^2-4ap\)
\(=4p\left(p-a\right)=VP\left(đpcm\right)\)
TC:a+b+cd=2p=>b+c=2p-a
=>(b+c)2=(2p-a)2
=>b2+2bc+c2=4p2-4pa+a2
=>b2+2bc+c2-a2=4p2-4pa
=>2bc+b2+c2-a2=4p(p-a) ĐPCM
\(2bc+b^2+c^2-a^2\)
\(=\left(b+c\right)^2-a^2\)
\(=\left(a+b+c\right)\left(b+c-a\right)\)
\(=2p\left(a+b+c-2a\right)\)
\(=2p\left(2p-2a\right)=4p\left(p-a\right)\)
biến đổi vế phải ta được:
4p(p -a ) = 4p\(^2\)-4pa
=(2p)\(^2\)-2p.2a
=(a+b+c)\(^2\)-2a(a+b+c)
=\(a^2+b^2+c^2+2ab+2ac+2bc\)-\(2a^2-2ab-2ac\)
=\(2bc+b^2+c^2-a^2\)=vế trái (đpcm)
a+b+c = 2p => 4p = 2(a+b+c); p=(a+b+c)/2
VP = 4p(p-a) = 2(a+b+c)(\(\frac{a+b+c}{2}-a\))
= \(2\left(a+b+c\right)\left(\frac{a+b+c-2a}{2}\right)\)
=\(2\left(a+b+c\right)\cdot\frac{b+c-a}{2}=\left(a+b+c\right)\left(b+c-a\right)\)
\(=ab+ac-a^2+b^2+bc-ab+bc+c^2-ac\)
\(=2bc+b^2+c^2-a^2\) = VT (đpcm)
4p(p-a)=2(a+b+c)[(b+c-a)/2]=(a+b+c)(c+b-a)(1)
b2+c2+2ab-a2=(a+b+c)(c+b-a)(2)
từ (1) và (2) suy ra b2+c2+2ab-a2=4p(p-a)