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a) \(A=\frac{x+2}{x+3}-\frac{5}{x^2+x-6}+\frac{1}{2-x}\left(ĐK:x\ne-3;x\ne2\right)\)
\(=\frac{x+2}{x+3}-\frac{5}{\left(x-2\right)\left(x+3\right)}-\frac{1}{x-2}\)
\(=\frac{\left(x+2\right)\left(x-2\right)-5-\left(x+3\right)}{\left(x-2\right)\left(x+3\right)}\)
\(=\frac{x^2-4-5-x-3}{\left(x-2\right)\left(x+3\right)}=\frac{x^2-x-12}{\left(x-2\right)\left(x+3\right)}=\frac{\left(x-4\right)\left(x+3\right)}{\left(x-2\right)\left(x+3\right)}=\frac{x-4}{x-2}\)
Để \(A=-\frac{3}{4}\)
\(\Leftrightarrow\frac{x-4}{x-2}=-\frac{3}{4}\)
\(\Leftrightarrow4\left(x-4\right)=-3\left(x-2\right)\)
\(\Leftrightarrow4x-16=-3x+6\)
\(\Leftrightarrow7x=22\Leftrightarrow x=\frac{22}{7}\left(tm\right)\)
Vậy \(x=\frac{22}{7}\) thì \(A=-\frac{3}{4}\)
b) \(A=\frac{x-4}{x-2}=\frac{\left(x-2\right)-2}{x-2}=1-\frac{2}{x-2}\)
Để \(A\in Z\Rightarrow\frac{2}{x-2}\in Z\Rightarrow x-2\inƯ\left(2\right)\)
Mà: \(Ư\left(2\right)=\left\{1;-1;2;-2\right\}\)
=> \(x-2\in\left\{1;-1;2;-2\right\}\)
+) \(x-2=1\Rightarrow x=3\left(tm\right)\)
+) \(x-2=-1\Rightarrow x=1\left(tm\right)\)
+) \(x-2=2\Rightarrow x=4\left(tm\right)\)
+) \(x-2=-2\Rightarrow x=0\left(tm\right)\)
Vậy \(x\in\left\{0;1;3;4\right\}\) thì \(A\in Z\)
A=x+2/x+3-5/(x-2)(x+3)-1/x-2
A=(x+2)(x-2)-5-x-3/(x-2)(x+3)
A=x^2-4-5-x-3/(x-2)(x+3)
A=x^2-x-12/(x-2)(x+3)
A=(x+3)(x-4)/(x-2)(x+3)
A=x-4/x-2
Để A=-3/4 thì x-4/x-2=-3/4
Từ đó suy ra (x-4)4=-3(x-2)
4x-16=-3x+6
7x=22
x=22/7
b,Do A nguyên nên x-4/x-2 nguyên(x#2)
suy ra x-4-x+2 chia hết cho x-2
nên 2 chia hết cho x-2
mà ước 2=-2;-1;1;2
nên x=0;1;3;4
1. P = \(\frac{x+2}{x+3}-\frac{5}{x^2+x-6}+\frac{1}{2-x}\) ĐKXĐ: \(x\ne-3\), \(x\ne2\)
= \(\frac{x+2}{x+3}-\frac{5}{\left(x+3\right)\left(x-2\right)}-\frac{1}{x-2}\)
= \(\frac{x^2-4}{\left(x+3\right)\left(x-2\right)}-\frac{5}{\left(x+3\right)\left(x-2\right)}-\frac{x+3}{x-2}\)
= \(\frac{x^2-4-5-x-3}{\left(x+3\right)\left(x-2\right)}\)
= \(\frac{x^2-x-12}{\left(x+3\right)\left(x-2\right)}\)
= \(\frac{\left(x-4\right)\left(x+3\right)}{\left(x+3\right)\left(x-2\right)}\)
= \(\frac{x-4}{x-2}\)
2. P=\(\frac{-3}{4}\)
<=> \(\frac{x-4}{x-2}=\frac{-3}{4}\)
<=> 4 ( x - 4 ) = -3 ( x - 2 )
<=> 4x - 16 = -3x + 6
<=> 7x = 2
<=> x = \(\frac{22}{7}\)
3. \(x^2-9=0\)
<=> ( x -3 ) ( x + 3 ) = 0
<=> \(\orbr{\begin{cases}x=3\left(tm\right)\\x=-3\left(ktm\right)\end{cases}}\)
-> P = \(\frac{3-4}{3-2}\) = -1
Bài 1.
a)\(\frac{4x-4}{x^2-4x+4}\div\frac{x^2-1}{\left(2-x\right)^2}=\frac{4\left(x-1\right)}{\left(x-2\right)^2}\div\frac{\left(x-1\right)\left(x+1\right)}{\left(x-2\right)^2}=\frac{4\left(x-1\right)}{\left(x-2\right)^2}\times\frac{\left(x-2\right)^2}{\left(x-1\right)\left(x+1\right)}=\frac{4}{x+1}\)
b) \(\frac{2x+1}{2x^2-x}+\frac{32x^2}{1-4x^2}+\frac{1-2x}{2x^2+x}=\frac{2x+1}{x\left(2x-1\right)}+\frac{-32x^2}{4x^2-1}+\frac{1-2x}{x\left(2x+1\right)}\)
\(=\frac{\left(2x+1\right)\left(2x+1\right)}{x\left(2x-1\right)\left(2x+1\right)}+\frac{-32x^3}{x\left(2x-1\right)\left(2x+1\right)}+\frac{\left(1-2x\right)\left(2x-1\right)}{x\left(2x-1\right)\left(2x+1\right)}\)
\(=\frac{4x^2+4x+1}{x\left(2x-1\right)\left(2x+1\right)}+\frac{-32x^3}{x\left(2x-1\right)\left(2x+1\right)}+\frac{-4x^2+4x-1}{x\left(2x-1\right)\left(2x+1\right)}\)
\(=\frac{4x^2+4x+1-32x^3-4x^2+4x-1}{x\left(2x-1\right)\left(2x+1\right)}=\frac{-32x^3+8x}{x\left(2x-1\right)\left(2x+1\right)}\)
\(=\frac{-8x\left(4x^2-1\right)}{x\left(2x-1\right)\left(2x+1\right)}=\frac{-8x\left(2x-1\right)\left(2x+1\right)}{x\left(2x-1\right)\left(2x+1\right)}=-8\)
c) \(\left(\frac{1}{x+1}+\frac{1}{x-1}-\frac{2x}{1-x^2}\right)\times\frac{x-1}{4x}\)
\(=\left(\frac{1}{x+1}+\frac{1}{x-1}+\frac{2x}{x^2-1}\right)\times\frac{x-1}{4x}\)
\(=\left(\frac{x-1}{\left(x-1\right)\left(x+1\right)}+\frac{x+1}{\left(x-1\right)\left(x+1\right)}+\frac{2x}{\left(x-1\right)\left(x+1\right)}\right)\times\frac{x-1}{4x}\)
\(=\left(\frac{x-1+x+1+2x}{\left(x-1\right)\left(x+1\right)}\right)\times\frac{x-1}{4x}\)
\(=\frac{4x}{\left(x-1\right)\left(x+1\right)}\times\frac{x-1}{4x}=\frac{1}{x+1}\)
Bài 3.
N = ( 4x + 3 )2 - 2x( x + 6 ) - 5( x - 2 )( x + 2 )
= 16x2 + 24x + 9 - 2x2 - 12x - 5( x2 - 4 )
= 14x2 + 12x + 9 - 5x2 + 20
= 9x2 + 12x + 29
= 9( x2 + 4/3x + 4/9 ) + 25
= 9( x + 2/3 )2 + 25 ≥ 25 > 0 ∀ x
=> đpcm
b, P=x+2x+3−5x2+3x−2x−6+12−xP=x+2x+3−5x2+3x−2x−6+12−x
=x+2x+3−5(x+3)(x−2)−1x−2=x+2x+3−5(x+3)(x−2)−1x−2
=(x+2)(x−2)(x+3)(x−2)−5(x+3)(x−2)−x+3(x+3)(x−2)=(x+2)(x−2)(x+3)(x−2)−5(x+3)(x−2)−x+3(x+3)(x−2)
=x2−4−5−x−3(x+3)(x−2)=x2−x−12(x+3)(x−2)=x2−4−5−x−3(x+3)(x−2)=x2−x−12(x+3)(x−2)
=x2−4x+3x−12(x+3)(x−2)=x2−4x+3x−12(x+3)(x−2)
=(x−4)(x+3)(x+3)(x−2)=x−4x−2=(x−4)(x+3)(x+3)(x−2)=x−4x−2
c, Để P=−34P=−34
⇔x−4x−2=−34⇔x−4x−2=−34
⇔4(x−4)=−3(x−2)⇔4(x−4)=−3(x−2)
⇔4x−16+3x−6=0⇔4x−16+3x−6=0
⇔7x−22=0⇔7x−22=0
⇔x=227⇔x=227
d, Để P có giá trị nguyên
⇔x−4⋮x−2⇔x−4⋮x−2
⇔(x−2)−2⋮x−2⇔(x−2)−2⋮x−2
⇔2⋮x−2⇔x−2∈Ư(2)={1;−1;2;−2}⇔2⋮x−2⇔x−2∈Ư(2)={1;−1;2;−2}
| x−2x−2 | 1 | -1 | 2 | -2 |
| x | 3 | 1 | 4 | 0 |
e,
x2−9=0x2−9=0
⇒x2=9⇒[x=3x=−3⇒x2=9⇒[x=3x=−3
Với x=3,có :
x−4x−2=3−43−2=−11=−1x−4x−2=3−43−2=−11=−1
Với x=-3,có :
x−4x−2=−3−4−3−2=75x−4x−2=−3−4−3−2=75
P = \(\frac{x+2}{x+3}-\frac{5}{x^2+x-6}+\frac{1}{2-x}\)
= \(1-\frac{1}{x+3}-\frac{5}{\left(x-2\right)\left(x+3\right)}-\frac{1}{x-2}\)
= \(1-\left(\frac{1}{x+3}+\frac{1}{x-2}+\frac{5}{\left(x-2\right)\left(x+3\right)}\right)\)
\(=1-\left(\frac{x+3+x-2+5}{\left(x-2\right)\left(x+3\right)}\right)=1-\frac{2x+6}{\left(x-2\right)\left(x+3\right)}=1-\frac{2\left(x+3\right)}{\left(x-2\right)\left(x+3\right)}\)
\(=1-\frac{2}{x-2}\)
Khi đó P = \(1-\frac{2}{x-2}\)
Khi P = -3/4
=> \(1-\frac{2}{x-2}=-\frac{3}{4}\)
=> \(\frac{2}{x-2}=\frac{7}{4}\)
=> 7(x - 2) = 2.4
=> 7(x - 2) = 8
=> x - 2 = 8/7
=> x = \(\frac{22}{7}\)
Vậy khi x = 22/7 thì P = -3/4
\(P=\frac{x+2}{x+3}-\frac{5}{x^2+x-6}+\frac{1}{2-x}\)
\(=\frac{x+2}{x+3}-\frac{5}{\left(x-2\right)\left(x+3\right)}+\frac{1}{2-x}\)
\(=\frac{x+2}{x+3}-\frac{5}{\left(x-2\right)\left(x+3\right)}-\frac{1}{x-2}\)
\(=\frac{\left(x+2\right)\left(x-2\right)}{\left(x+3\right)\left(x-2\right)}-\frac{5}{\left(x-2\right)\left(x+3\right)}-\frac{x+3}{\left(x-2\right)\left(x+3\right)}\)
\(=\frac{x^2-4-5-x-3}{\left(x+3\right)\left(x-2\right)}=\frac{x^2-12-x}{\left(x+3\right)\left(x-2\right)}=\frac{\left(x-4\right)\left(x+3\right)}{\left(x+3\right)\left(x-2\right)}=\frac{x-4}{x-2}\)
Neu P = -3/4 thi :
\(\frac{x-4}{x-2}=-\frac{3}{4}\Leftrightarrow4x-16=-3x+6\Leftrightarrow7x=22\Leftrightarrow x=\frac{22}{7}\)