(x2 + 5x + 6) (x2 - 11x + 30) = 180
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12 tháng 2 2023
\(\Leftrightarrow\dfrac{1}{\left(x+2\right)\left(x+3\right)}+\dfrac{1}{\left(x+3\right)\left(x+4\right)}+...+\dfrac{1}{\left(x+5\right)\left(x+6\right)}=\dfrac{1}{8}\)
=>\(\dfrac{1}{x+2}-\dfrac{1}{x+3}+\dfrac{1}{x+3}-\dfrac{1}{x+4}+...+\dfrac{1}{x+5}-\dfrac{1}{x+6}=\dfrac{1}{8}\)
=>1/x+2-1/x+6=1/8
=>\(\dfrac{x+6-x-2}{\left(x+2\right)\left(x+6\right)}=\dfrac{1}{8}\)
=>x^2+8x+12=32
=>x^2+8x-20=0
=>(x+10)(x-2)=0
=>x=-10 hoặc x=2
8 tháng 1 2021
a) Ta có: \(x^3+x^2+x+1=0\)
\(\Leftrightarrow x^2\left(x+1\right)+\left(x+1\right)=0\)
\(\Leftrightarrow\left(x+1\right)\left(x^2+1\right)=0\)
mà \(x^2+1>0\forall x\)
nên x+1=0
hay x=-1
Vậy: S={-1}
b) Ta có: \(x^3-6x^2+11x-6=0\)
\(\Leftrightarrow x^3-x^2-5x^2+5x+6x-6=0\)
\(\Leftrightarrow x^2\left(x-1\right)-5x\left(x-1\right)+6\left(x-1\right)=0\)
\(\Leftrightarrow\left(x-1\right)\left(x^2-5x+6\right)=0\)
\(\Leftrightarrow\left(x-1\right)\left(x-2\right)\left(x-3\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x-1=0\\x-2=0\\x-3=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=1\\x=2\\x=3\end{matrix}\right.\)
Vậy: S={1;2;3}
c) Ta có: \(x^3-x^2-21x+45=0\)
\(\Leftrightarrow x^3-3x^2+2x^2-6x-15x+45=0\)
\(\Leftrightarrow x^2\left(x-3\right)+2x\left(x-3\right)-15\left(x-3\right)=0\)
\(\Leftrightarrow\left(x-3\right)\left(x^2+2x-15\right)=0\)
\(\Leftrightarrow\left(x-3\right)\left(x^2+5x-3x-15\right)=0\)
\(\Leftrightarrow\left(x-3\right)^2\cdot\left(x+5\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x-3=0\\x+5=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=3\\x=-5\end{matrix}\right.\)
Vậy: S={3;-5}
d) Ta có: \(x^4+2x^3-4x^2-5x-6=0\)
\(\Leftrightarrow x^4-2x^3+4x^3-8x^2+4x^2-8x+3x-6=0\)
\(\Leftrightarrow x^3\left(x-2\right)+4x^2\cdot\left(x-2\right)+4x\left(x-2\right)+3\left(x-2\right)=0\)
\(\Leftrightarrow\left(x-2\right)\left(x^3+4x^2+4x+3\right)=0\)
\(\Leftrightarrow\left(x-2\right)\left(x^3+3x^2+x^2+4x+3\right)=0\)
\(\Leftrightarrow\left(x-2\right)\left[x^2\left(x+3\right)+\left(x+1\right)\left(x+3\right)\right]=0\)
\(\Leftrightarrow\left(x-2\right)\left(x+3\right)\left(x^2+x+1\right)=0\)
mà \(x^2+x+1>0\forall x\)
nên (x-2)(x+3)=0
\(\Leftrightarrow\left[{}\begin{matrix}x-2=0\\x+3=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=2\\x=-3\end{matrix}\right.\)
Vậy: S={2;-3}
12 tháng 5 2023
c: =>(x+2)(x+3)(x-5)(x-6)=180
=>(x^2-3x-10)(x^2-3x-18)=180
=>(x^2-3x)^2-28(x^2-3x)=0
=>x(x-3)(x-7)(x+4)=0
=>\(x\in\left\{0;3;7;-4\right\}\)
c: =>(x-3)(x+2)(2x+1)(3x-1)=0
=>\(x\in\left\{3;-2;-\dfrac{1}{2};\dfrac{1}{3}\right\}\)
CT
1
NV
Nguyễn Việt Lâm
Giáo viên
1 tháng 3 2023
ĐKXĐ: \(x\ne\left\{-4;-5;-6;-7\right\}\)
\(\dfrac{1}{x^2+9x+20}+\dfrac{1}{x^2+11x+30}+\dfrac{1}{x^2+13x+42}=\dfrac{1}{18}\)
\(\Leftrightarrow\dfrac{1}{\left(x+4\right)\left(x+5\right)}+\dfrac{1}{\left(x+5\right)\left(x+6\right)}+\dfrac{1}{\left(x+6\right)\left(x+7\right)}=\dfrac{1}{18}\)
\(\Leftrightarrow\dfrac{1}{x+4}-\dfrac{1}{x+5}+\dfrac{1}{x+5}-\dfrac{1}{x+6}+\dfrac{1}{x+6}-\dfrac{1}{x+7}=\dfrac{1}{18}\)
\(\Leftrightarrow\dfrac{1}{x+4}-\dfrac{1}{x+7}=\dfrac{1}{18}\)
\(\Leftrightarrow\dfrac{3}{\left(x+4\right)\left(x+7\right)}=\dfrac{1}{18}\)
\(\Rightarrow\left(x+4\right)\left(x+7\right)=54\)
\(\Leftrightarrow x^2+11x-26=0\)
\(\Leftrightarrow x^2-2x+13x-26=0\)
\(\Leftrightarrow x\left(x-2\right)+13\left(x-2\right)=0\)
\(\Leftrightarrow\left(x-2\right)\left(x+13\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x=2\\x=-13\end{matrix}\right.\)
CM
19 tháng 1 2017
a) (x - 2)(x - 3). b) 3(x - 2)(x + 5).
c) (x - 2)(3x + 1). d) (x-2y)(x - 5y).
e) (x + l)(x + 2)(x - 3). g) (x-1)(x + 3)( x 2 + 3).
h) (x + y - 3)(x - y + 1).
VT
1
AH
Akai Haruma
Giáo viên
5 tháng 4 2021
Lời giải:
$(x^2+x)(x^2+11x+30)+7=x(x+1)(x+5)(x+6)+7$
$=(x^2+6x)(x^2+6x+5)+7$
$=(x^2+6x)^2+5(x^2+6x)+7$
$=(x^2+6x+\frac{5}{2})^2+\frac{3}{4}\geq \frac{3}{4}$ với mọi $x\in\mathbb{R}$
Do đó $\frac{3}{4}\geq k$ nên $k_{\max}=\frac{3}{4}$
HN
1
10 tháng 3 2022
\(f\left(x\right)=\dfrac{11x+3}{-x^2+5x-7}.\)
Ta có: \(-x^2+5x-7\) là 1 tam thức bậc 2.
\(\left\{{}\begin{matrix}a=-1< 0.\\\Delta=5^2-4.\left(-1\right).\left(-7\right)=-3< 0.\end{matrix}\right.\)
\(\Rightarrow-x^2+5x-7>0\forall x\in R.\)
\(\Rightarrow\) \(f\left(x\right)>0.\Leftrightarrow11x+3>0.\Leftrightarrow x>\dfrac{-3}{11}.\\ f\left(x\right)< 0.\Leftrightarrow11x+3>0.\Leftrightarrow x>\dfrac{-3}{11}.\\ f\left(x\right)=0.\Leftrightarrow x=\dfrac{-3}{11}.\)
NT
1
28 tháng 2 2022
\(a,x^2-11x+30=0\\ \Leftrightarrow x^2-5x-6x+30=0\\ \Leftrightarrow x\left(x-5\right)-6\left(x-5\right)=0\\ \Leftrightarrow\left(x-5\right)\left(x-6\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}x=5\\x=6\end{matrix}\right.\)
\(b,\Delta=\left(-8\right)^2-4.3\left(-5\right)=64+60=124\)
\(x_1=\dfrac{8+\sqrt{124}}{2.3}=\dfrac{8+2\sqrt{31}}{6}=\dfrac{4+\sqrt{31}}{3}\)
\(x_1=\dfrac{8-\sqrt{124}}{2.3}=\dfrac{8-2\sqrt{31}}{6}=\dfrac{4-\sqrt{31}}{3}\)
\(\left(x^2+5x+6\right)\left(x^2-11x+3x\right)=180\\ \Leftrightarrow\left(x+2\right)\left(x+3\right)\left(x-5\right)\left(x-6\right)=180\\ \Leftrightarrow\left[\left(x+2\right)\left(x-5\right)\right]\left[\left(x+3\right)\left(x-6\right)\right]\\ \Leftrightarrow\left(x^2-3x-10\right)\left(x^2-3x-18\right)=0\left(1\right)\)
Đặt \(x^2-3x-14=a\)
\(\left(1\right)\Leftrightarrow\left(a+4\right)\left(a-4\right)=180\\ \Leftrightarrow a^2-16=180\\ \Leftrightarrow a^2=196\\ \Leftrightarrow\left[{}\begin{matrix}a=14\\a=-14\end{matrix}\right.\\ \Leftrightarrow\left[{}\begin{matrix}x^2-3x-14=14\\x^2-3x-14=-14\end{matrix}\right.\\ \Leftrightarrow\left[{}\begin{matrix}x^2-3x=0\left(vô.lí\right)\\x^2-3x=0\end{matrix}\right.\\ \Leftrightarrow\left[{}\begin{matrix}x=0\\x=3\end{matrix}\right.\)