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\(\left(a+b+c\right)^2=a^2+b^2+c^2 \Leftrightarrow a^2+b^2+c^2+2ab+2ac+2bc=a^2+b^2+c^2\)
<=> \(ab+bc+ac=0\Leftrightarrow\frac{ab+ac+bc}{abc}=0\Leftrightarrow\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0\Leftrightarrow\frac{1}{a}+\frac{1}{b}=-\frac{1}{c}\)
<=> \(\left(\frac{1}{a}+\frac{1}{b}\right)^3=\frac{1}{c^3}\Leftrightarrow\frac{1}{a^3}+\frac{1}{b^3}+3.\frac{1}{a^2}.\frac{1}{b}+3.\frac{1}{a}.\frac{1}{b^2}=-\frac{1}{c^3}\)
\(\Leftrightarrow\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}+\frac{3}{ab}\left(\frac{1}{a}+\frac{1}{b}\right)=0\Leftrightarrow\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}+\frac{3}{ab}\left(\frac{-1}{c}\right)=0\Leftrightarrow\)dpcm
1. A = -4 phần x+2
2. 2x^2 + x = 0 => x = 0 hoặc x = -1/2
Với x = 0 thì A = -2
Với x = -1/2 thì A = -8/3
3. A = 1/2 => -4 phần x + 2 = 1/2
<=> -8 = x + 2
<=> x = -10
4. A nguyên dương => A > 0
=> -4 phần x + 2 > 0
Do -4 < 0 nên -4 phần x + 2 > 0 khi x + 2 < 0
=> x < -2
1) \(Q=\frac{x^2-2x-1}{x^2}=1-\frac{2}{x}-\frac{1}{x^2}\). Đặt \(y=\frac{1}{x}\), ta có :
\(Q=-y^2-2y+1=-\left(y^2+2y+1\right)+2=-\left(y+1\right)^2+2\le2\)
Dấu "=" xảy ra \(\Leftrightarrow y=-1\Leftrightarrow\frac{1}{x}=-1\Leftrightarrow x=-1\)
Vậy Max Q = 2 tại x = -1
2) \(1-9x^2=\left(1-3x\right)\left(1+3x\right)\)
3) \(\frac{x^2}{9}-\frac{y^2}{16}=\left(\frac{x}{3}-\frac{y}{4}\right)\left(\frac{x}{3}+\frac{y}{4}\right)\)
4) \(a^4-b^4=\left(a^2-b^2\right)\left(a^2+b^2\right)=\left(a-b\right)\left(a+b\right)\left(a^2+b^2\right)\)
5) \(\left(a-b\right)^2-1=\left(a-b+1\right)\left(a-b-1\right)\)
6) \(4-\left(a-b\right)^2=\left(2-a+b\right)\left(2+a-b\right)\)
7) \(\left(x-y\right)^2-\left(m+n\right)^2=\left(x-y-m-n\right)\left(x-y+m+n\right)\)
8) \(\left(3x-2y\right)^2-\left(2x-3y\right)^2=\left(3x-2y-2x+3y\right)\left(3x-2y+2x-3y\right)\)
\(=\left[3\left(x+y\right)-2\left(x+y\right)\right]\left[3\left(x-y\right)+2\left(x-y\right)\right]=5\left(x+y\right)\left(x-y\right)\)
9) \(4x^2-12xy+9y^2=\left(2x-3y\right)^2\)
10) \(\left(x^4+2x^2+1\right)=\left(x^2+1\right)^2\)
11) \(\left(a^4+4-4x^2\right)=\left(a^2-2\right)^2\)
1. x2-4x+4+9=(x-4x+4)+9=(x-2)2+9 >=9. nên pt vô nghiệm
2. \(a+b\ge2\sqrt{ab}\Leftrightarrow\left(a+b\right)^2\ge4ab\Leftrightarrow a^2+2ab+b^2\ge4ab\)
\(\Leftrightarrow a^2-2ab+b^2\ge0\Leftrightarrow\left(a-b\right)^2\ge0\)( đúng). dpcm