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Đặt \(A=\frac{x^2+x-6}{x^3-4x^2-18x+9}\)
\(A=\frac{x^2+3x-2x-6}{x^3+3x^2-7x^2-21x+3x+9}\)
\(A=\frac{x\left(x+3\right)-2\left(x+3\right)}{x^2\left(x+3\right)-7x\left(x+3\right)+3\left(x+3\right)}\)
\(A=\frac{\left(x-2\right)\left(x+3\right)}{\left(x^2-7x+3\right)\left(x+3\right)}\)
\(A=\frac{x-2}{x^2-7x+3}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
Ta có :
\(\dfrac{x^2+x-6}{x^3-4x^2-18x+9}=\dfrac{\left(x-2\right)\left(x+3\right)}{\left(x+3\right)\left(x^2-7x+3\right)}=\dfrac{x-2}{x^2-7x+3}\)
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a: ĐKXĐ: \(x\notin\left\{-3;3\right\}\)
b: \(P=\dfrac{x^2-2x-3-x^2-5x-6+4x+6}{\left(x-3\right)\left(x+3\right)}\)
\(=\dfrac{-3x-3}{\left(x-3\right)\left(x+3\right)}\)
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\(A=\frac{4x}{x^2-2x}+\frac{3}{2-x}+\frac{12x}{x^3-4x}\)
\(A=\frac{4x}{x\left(x-2\right)}-\frac{3}{x-2}+\frac{12x}{x\left(x-2\right)\left(x+2\right)}\)
\(A=\frac{4x\left(x+2\right)-3x\left(x+2\right)+12x}{x\left(x-2\right)\left(x+2\right)}\)
\(A=\frac{x\left(x+2\right)+12x}{x\left(x-2\right)\left(x+2\right)}\)
\(A=\frac{x^2+2x+12x}{x\left(x-2\right)\left(x+2\right)}\)
\(A=\frac{x^2+14x}{x\left(x-2\right)\left(x+2\right)}\)
\(\frac{x^2+x-6}{x^3-4x^2-18x+9}=\frac{x^2+3x-2x-6}{x^3+3x^2-7x^2-21x+3x+9}\)
\(=\frac{x\left(x+3\right)-2\left(x+3\right)}{x^2\left(x+3\right)-7x\left(x+3\right)+3\left(x+3\right)}\)
\(=\frac{\left(x+3\right)\left(x-2\right)}{\left(x+3\right)\left(x^2-7x+3\right)}=\frac{x-2}{x^2-7x+3}\) (điều kiện: x khác -3)
t phân tích \(x^2-7x+3\) được như này =))
\(x^2-7x+3=x^2-2.x.\frac{7}{2}+\left(\frac{7}{2}\right)^2-\frac{49}{4}+3\)
\(=\left(x-\frac{7}{2}\right)^2-\frac{37}{4}\)
\(=\left(x-\frac{7}{2}\right)^2-\left(\frac{\sqrt{37}}{2}\right)^2\)
\(=\left(x-\frac{7}{2}-\frac{\sqrt{37}}{2}\right)\left(x-\frac{7}{2}+\frac{\sqrt{37}}{2}\right)\)
\(=\left(x-\frac{7+\sqrt{37}}{2}\right)\left(x-\frac{7-\sqrt{37}}{2}\right)\)