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ND
25 tháng 2 2018
a.
\(x\left(x-1\right)\left(x+1\right)\left(x+2\right)=24\)
\(\Leftrightarrow x\left(x+1\right).\left(x-1\right)\left(x+2\right)-24=0\)
\(\Leftrightarrow\left(x^2+x\right)\left(x^2+x-2\right)-24=0\)
Đặt \(a=x^2+x-1\) , ta có pt:
\(\left(a+1\right)\left(a-1\right)-24=0\)
\(\Leftrightarrow a^2-1-24=0\)
\(\Leftrightarrow a^2-25=0\)
\(\Leftrightarrow\left(a-5\right)\left(a+5\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}a=5\\a=-5\end{matrix}\right.\)
*Với a = 5 ta được:
\(x^2+x-1=5\)
\(\Leftrightarrow x^2+x-6=0\)
\(\Leftrightarrow x^2+3x-2x-6=0\)
\(\Leftrightarrow\left(x^2+3x\right)-\left(2x+6\right)=0\)
\(\Leftrightarrow x\left(x+3\right)-2\left(x+3\right)=0\)
\(\Leftrightarrow\left(x+3\right)\left(x-2\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-3\\x=2\end{matrix}\right.\)
*Với a = -5 ta được:
\(x^2+x-1=-5\)
\(\Leftrightarrow x^2+x+4=0\)
\(\Leftrightarrow x^2+2.x.\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{15}{4}=0\)
\(\Leftrightarrow\left(x+\dfrac{1}{2}\right)^2+\dfrac{15}{4}=0\) ( loại)
Vậy pt có tập nghiệm là: \(s=\left\{-3;2\right\}\)
25 tháng 2 2018
c)(ĐKXĐ: x khác 30;29)
\(\Leftrightarrow\dfrac{x-29}{30}-1+\dfrac{x-30}{29}-1=\dfrac{29}{x-30}-1+\dfrac{30}{x-29}-1\)
\(\Leftrightarrow\dfrac{x-59}{30}+\dfrac{x-59}{29}=\dfrac{x-59}{30-x}+\dfrac{x-59}{29-x}\)
\(\Leftrightarrow x=59\)(tm) or \(\dfrac{1}{30}+\dfrac{1}{29}-\dfrac{1}{30-x}-\dfrac{1}{29-x}=0\)
\(\Leftrightarrow\dfrac{-x}{30\left(30-x\right)}+\dfrac{-x}{29\left(29-x\right)}=0\)
\(\Leftrightarrow x=0\)(tm) or \(\dfrac{1}{30\left(30-x\right)}+\dfrac{1}{29\left(29-x\right)}=0\)
\(\Leftrightarrow1741-59x=0\)
\(\Leftrightarrow x=\dfrac{1741}{59}\left(tm\right)\)
Vậy S={0;\(\dfrac{1741}{59}\);59}
31 tháng 1 2020
Câu c : \(x^4-3x^3+2x^2-9x+9=0\)
<=>\(x^4-x^3-2x^3+2x^2-9x+9=0\)
<=>\(x^3\left(x-1\right)-2x^2\left(x-1\right)-9\left(x-1\right)=0\)
<=>\(\left(x-1\right)\left(x^3-2x^2-9\right)=0\)
<=> \(x-1=0\) hoặc \(x^3-2x^2-9=0\)
Nếu x-1=0 <=> x=1
Nếu \(x^3-2x^2-9=0\)
<=> \(x^3-3x^2+x^2-9=0\)
<=>\(x^2\left(x-3\right)+\left(x-3\right)\left(x+3\right)=0\)
<=>\(\left(x-3\right)\left(x^2+x+3\right)=0\)
Vì \(x^2+x+3=\left(x+\frac{1}{2}\right)^2+\frac{11}{4}\) >0 nên x-3=0 <=> x=3
Vậy \(S=\left\{1;3\right\}\)
31 tháng 1 2020
Câu b : \(x^2+\left(\frac{x}{x+1}\right)^2=\frac{5}{4}\)
<=> \(4x^2\left(x^2+2x+2\right)=5\left(x^2+2x+1\right)\)
<=> \(4x^4+8x^3+8x^2=5x^2+10x+5\)
<=>\(4x^4+8x^3+3x^2-10x-5=0\)
<=>\(4x^4-4x^3+12x^3-12x^2+15x^2-15x+5x-5=0\)
<=>\(\left(x-1\right)\left(4x^3+12x^2+15x+5\right)=0\)
<=>\(\left(x-1\right)\left(2x+1\right)\left(2x^2+5x+5\right)=0\)
<=>x=1 hoặc \(x=\frac{-1}{2}\)
Phương trình \(2x^2+5x+5=0\) Vô nghiệm
NT
1
PA
30 tháng 7 2017
(x2 + 5x + 6)(x2 + 9x + 20) = 24
<=> (x + 2)(x + 3)(x + 4)(x + 5) - 24 = 0
<=> (x2 + 7x + 10)(x2 + 7x + 12) - 24 = 0 (1)
Đặt x2 + 7x + 11 = t, ta có:
(1) <=> (t - 1)(t + 1) - 24 = 0
<=> t2 - 1 - 24 = 0
<=> (t - 5)(t + 5) = 0
\(\Leftrightarrow\left[{}\begin{matrix}t-5=0\\t+5=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x^2+7x+11-5=0\\x^2+7x+11+5=0\end{matrix}\right.\)
<=> (x + 1)(x + 6) = 0 (vì \(x^2+7x+16\ge\dfrac{15}{4}>0\))
<=> x = - 1 hoặc x = - 6
~ ~ ~ ~ ~
x4 - 24x = 32
<=> x4 - 24x - 32 = 0
<=> (x2 - 2x - 4)(x2 + 2x + 8) = 0
<=> \(\left(x-1-\sqrt{5}\right)\left(x-1+\sqrt{5}\right)=0\) (vì \(x^2+2x+8\ge7>0\))
\(\Leftrightarrow\left[{}\begin{matrix}x=1+\sqrt{5}\\x=1-\sqrt{5}\end{matrix}\right.\)
MN
31 tháng 1 2020
a) \(ĐKXĐ:x\ne-1;x\ne-3;x\ne-8;x\ne-10\)
\(\frac{2}{x^2+4x+3}+\frac{5}{x^2+11x+24}+\frac{2}{x^2+18x+8x}=\frac{9}{52}\)
\(\Leftrightarrow\frac{2}{\left(x+1\right)\left(x+3\right)}+\frac{5}{\left(x+3\right)\left(x+8\right)}+\frac{2}{\left(x+10\right)\left(x+8\right)}-\frac{9}{52}=0\)
\(\Leftrightarrow\frac{104\left(x+10\right)\left(x+8\right)+260\left(x+1\right)\left(x+10\right)+104\left(x+1\right)\left(x+3\right)-9\left(x+1\right)\left(x+3\right)\left(x+8\right)\left(x+10\right)}{52\left(x+1\right)\left(x+3\right)\left(x+8\right)\left(x+10\right)}=0\)
Đoạn này cậu tự phân tích tử rồi rút gọn nhé :D Vì hơi dài nên viết ra đây sẽ rối, k đẹp mắt cho lắm :>
\(\Leftrightarrow\frac{-927x^2+1782x+9072-9x^4-198x^3}{52\left(x+1\right)\left(x+3\right)\left(x+8\right)\left(x+10\right)}=0\)
\(\Leftrightarrow\frac{-9\left(x^4+22x^3+103x^2-198x-1008\right)}{52\left(x+1\right)\left(x+3\right)\left(x+8\right)\left(x+10\right)}=0\)
\(\Leftrightarrow\frac{-9\left(x^4-3x^3+25x^3-75x^{^2}+178x^2-534x+336x-1008\right)}{52\left(x+1\right)\left(x+3\right)\left(x+8\right)\left(x+10\right)}=0\)
\(\Leftrightarrow\frac{-9\left[x^3\left(x-3\right)+25x^2\left(x-3\right)+178x\left(x-3\right)+336\left(x-3\right)\right]}{52\left(x+1\right)\left(x+3\right)\left(x+8\right)\left(x+10\right)}=0\)
\(\Leftrightarrow\frac{-9\left(x-3\right)\left(x^3+25x^2+178x+336\right)}{52\left(x+1\right)\left(x+3\right)\left(x+8\right)\left(x+10\right)}=0\)
\(\Leftrightarrow\frac{-9\left(x-3\right)\left(x^3+14x^2+11x^2+154x+24x+336\right)}{52\left(x+1\right)\left(x+3\right)\left(x+8\right)\left(x+10\right)}=0\)
\(\Leftrightarrow\frac{-9\left(x-3\right)\left[x^2\left(x+14\right)+11x\left(x+14\right)+24\left(x+14\right)\right]}{52\left(x+1\right)\left(x+3\right)\left(x+8\right)\left(x+10\right)}=0\)
\(\Leftrightarrow\frac{-9\left(x-3\right)\left(x+14\right)\left(x^2+11x+24\right)}{52\left(x+1\right)\left(x+3\right)\left(x+8\right)\left(x+10\right)=0}\)
\(\Leftrightarrow\frac{-9\left(x+14\right)\left(x-3\right)\left(x+3\right)\left(x+8\right)}{52\left(x+1\right)\left(x+3\right)\left(x+8\right)\left(x+10\right)}=0\)
\(\Leftrightarrow\frac{-9\left(x+14\right)\left(x-3\right)}{52\left(x+1\right)\left(x+10\right)}=0\)
\(\Leftrightarrow-9x^2-99x+378=0\)
\(\Leftrightarrow x^2+11x-42=0\)
\(\Leftrightarrow\left(x+14\right)\left(x-3\right)=0\)
\(\Leftrightarrow\orbr{\begin{cases}x+14=0\\x-3=0\end{cases}\Leftrightarrow\orbr{\begin{cases}x=-14\\x=3\end{cases}}}\)
Vậy tập nghiệm của phương trình là : \(S=\left\{-14;3\right\}\)
b) \(ĐKXĐ:x\ne-1\)
\(x^2+\left(\frac{x}{x+1}\right)^2=\frac{5}{4}\)
\(\Leftrightarrow x^2+\frac{x^2}{\left(x+1\right)^2}-\frac{5}{4}=0\)
\(\Leftrightarrow\frac{4x^2\left(x^2+2x+1\right)+4x^2-5\left(x^2+2x+1\right)}{\left(x+1\right)^2}=0\)
\(\Leftrightarrow4x^4+8x^3+4x^2+4x^2-5x^2-10x-5=0\)
\(\Leftrightarrow4x^2+8x^3+3x^2-10x-5=0\)
\(\Leftrightarrow4x^4+2x^3+6x^3+3x^2-10x-5=0\)
\(\Leftrightarrow2x^3\left(2x+1\right)+3x^2\left(2x+1\right)-5\left(2x+1\right)=0\)
\(\Leftrightarrow\left(2x+1\right)\left(2x^3+3x^2-5\right)=0\)
\(\Leftrightarrow\left(2x+1\right)\left(2x^3-2x^2+5x^2-5x+5x-5\right)=0\)
\(\Leftrightarrow\left(2x+1\right)\left[2x^2\left(x-1\right)+5x\left(x-1\right)+5\left(x-1\right)\right]=0\)
\(\Leftrightarrow\left(2x+1\right)\left(x-1\right)\left(2x^2+5x+5\right)=0\)
\(\Leftrightarrow2x+1=0\)
hoặc \(x-1=0\)
hoặc \(2x^2+5x+5=0\)
\(\Leftrightarrow\) \(x=-\frac{1}{2}\left(tm\right)\)
hoặc \(x=1\left(tm\right)\)
hoặc \(\left(x+\frac{5}{4}\right)^2+\frac{55}{16}=0\left(ktm\right)\)
Vậy tập nghiệm của phương trình là : \(S=\left\{-\frac{1}{2};1\right\}\)
c) \(x^4-3x^3+2x^2-9x+9=0\)
\(\Leftrightarrow x^4-x^3-2x^3+2x^2-9x+9=0\)
\(\Leftrightarrow x^3\left(x-1\right)-2x^2\left(x-1\right)-9\left(x-1\right)=0\)
\(\Leftrightarrow\left(x-1\right)\left(x^3-2x^2-9\right)=0\)
\(\Leftrightarrow\left(x-1\right)\left[\left(x^3-3x^2\right)+\left(x^2-9\right)\right]=0\)
\(\Leftrightarrow\left(x-1\right)\left[x^2\left(x-3\right)+\left(x-3\right)\left(x+3\right)\right]=0\)
\(\Leftrightarrow\left(x-1\right)\left(x-3\right)\left(x^2+x+3\right)=0\)
\(\Leftrightarrow\)\(x-1=0\)
hoặc \(x-3=0\)
hoặc \(x^2+x+3=0\)
\(\Leftrightarrow\)\(x=1\left(tm\right)\)
hoặc \(x=3\left(tm\right)\)
hoặc \(\left(x-\frac{1}{2}\right)^2+\frac{11}{4}=0\left(ktm\right)\)
Vậy tập nghiệm của phương trình là :\(S=\left\{1;3\right\}\)
KN
3 tháng 2 2020
\(ĐKXĐ:x\ne-1;x\ne-3;x\ne-8;x\ne-10\)
\(pt\Leftrightarrow\frac{2}{\left(x+1\right)\left(x+3\right)}+\frac{5}{\left(x+3\right)\left(x+8\right)}+\frac{2}{\left(x+8\right)\left(x+10\right)}=\frac{9}{52}\)
\(\Leftrightarrow\frac{\left(x+3\right)-\left(x+1\right)}{\left(x+1\right)\left(x+3\right)}+\frac{\left(x+8\right)-\left(x+3\right)}{\left(x+3\right)\left(x+8\right)}+\frac{\left(x+10\right)-\left(x+8\right)}{\left(x+8\right)\left(x+10\right)}\)
\(=\frac{9}{52}\)
\(\Leftrightarrow\frac{1}{x+1}-\frac{1}{x+3}+\frac{1}{x+3}-\frac{1}{x+8}+\frac{1}{x+8}-\frac{1}{x+10}=\frac{9}{52}\)
\(\Leftrightarrow\frac{1}{x+1}-\frac{1}{x+10}=\frac{9}{52}\)
\(\Leftrightarrow\frac{9}{\left(x+1\right)\left(x+10\right)}=\frac{9}{52}\)
\(\Leftrightarrow\left(x+1\right)\left(x+10\right)=52\)
\(\Leftrightarrow x^2+11x+10=52\)
\(\Leftrightarrow x^2+11x-42=0\)
\(\Delta=11^2+4.42=289,\sqrt{289}=17\)
\(\Rightarrow\orbr{\begin{cases}x=\frac{-11+17}{2}=3\\x=\frac{-11-17}{2}=-14\end{cases}}\)
Vậy pt có 2 nghiệm là 3 và -14
TN
1
4 tháng 4 2020
a) x^4 - 5x^2 + 4 = 0
<=> (x^2 - 1)(x^2 - 4) = 0
<=> x^2 - 1 = 0 hoặc x^2 - 4 = 0
<=> x = +-1 hoặc x = +-2
b) x^4 - 10x^2 + 9 = 0
<=> (x^2 - 1)(x^2 - 9) = 0
<=> x^2 - 1 = 0 hoặc x^2 - 9 = 0
<=> x = +-1 hoặc x = +-3
c) x^3 + 6x^2 + 11x + 6 = 0
<=> (x^2 + 5x + 6)(x + 1) = 0
<=> (x + 2)(x + 3)(x + 1) = 0
<=> x + 2 = 0 hoặc x + 3 = 0 hoặc x + 1 = 0
<=> x = -2 hoặc x = -3 hoặc x = -1
d) x^3 + 9x^2 + 26x + 24 = 0
<=> (x^2 + 7x + 12)(x + 2) = 0
<=> (x + 3)(x + 4)(x + 2) = 0
<=> x + 3 = 0 hoặc x + 4 = 0 hoặc x + 2 = 0
<=> x = -3 hoặc x = -4 hoặc x = -2
16 tháng 6 2022
24:
\(\Leftrightarrow\dfrac{1}{\left(x+2\right)\left(x+3\right)}+\dfrac{1}{\left(x+3\right)\left(x+4\right)}+\dfrac{1}{\left(x+4\right)\left(x+5\right)}+\dfrac{1}{\left(x+5\right)\left(x+6\right)}=\dfrac{1}{8}\)
\(\Leftrightarrow\dfrac{1}{x+2}-\dfrac{1}{x+6}=\dfrac{1}{8}\)
\(\Leftrightarrow\left(x+2\right)\left(x+6\right)=8\left(x+6\right)-8\left(x+2\right)\)
\(\Leftrightarrow x^2+8x+12=8x+48-8x-16=32\)
=>(x+10)(x-2)=0
=>x=-10 hoặc x=2
25: \(\Leftrightarrow\dfrac{\left(x+1\right)^2+1}{x+1}+\dfrac{\left(x+4\right)^2+4}{x+4}=\dfrac{\left(x+2\right)^2+2}{x+2}+\dfrac{\left(x+3\right)^2+3}{x+3}\)
\(\Leftrightarrow x+1+\dfrac{1}{x+1}+x+4+\dfrac{4}{x+4}=x+2+\dfrac{2}{x+2}+x+3+\dfrac{3}{x+3}\)
\(\Leftrightarrow\dfrac{1}{x+1}+\dfrac{4}{x+4}=\dfrac{2}{x+2}+\dfrac{3}{x+3}\)
\(\Leftrightarrow x+5=0\)
hay x=-5
Trả lời:
(bài này tìm GTNN đúng không?)
a, \(x^2+11x+24=x^2+2.x.\frac{11}{2}+\frac{121}{4}-\frac{25}{4}=\left(x+\frac{11}{2}\right)^2-\frac{25}{4}\ge-\frac{25}{4}\forall x\)
Dấu "=" xảy ra khi x + 11/2 = x = - 11/2
Vậy GTNN của bt = - 25/4 khi x = - 11/2
b, \(x^2+x-24=x^2+2.x.\frac{1}{2}+\frac{1}{4}-\frac{97}{4}=\left(x+\frac{1}{2}\right)^2-\frac{97}{4}\ge-\frac{97}{4}\ge-\frac{97}{4}\forall x\)
Dấu "=" xảy ra khi x + 1/2 = 0 <=> x = - 1/2
Vậy GTNN của bt = - 97/4 khi x = - 1/2
c, \(x^2+9x+20=x^2+2.x.\frac{9}{2}+\frac{81}{4}-\frac{1}{4}=\left(x+\frac{9}{2}\right)^2-\frac{1}{4}\ge-\frac{1}{4}\forall x\)
Dấu "=" xảy ra khi x + 9/2 = 0 <=> x = - 9/2
Vậy GTNN của bt = - 1/4 khi x = - 9/2