Hãy nhập câu hỏi của bạn vào đây, nếu là tài khoản VIP, bạn sẽ được ưu tiên trả lời.
a, - Đặt \(x^2+x=a\) ta được phương trình :\(a^2+4a-12=0\)
=> \(a^2-2a+6a-12=0\)
=> \(a\left(a-2\right)+6\left(a-2\right)=0\)
=> \(\left(a+6\right)\left(a-2\right)=0\)
=> \(\left[{}\begin{matrix}a+6=0\\a-2=0\end{matrix}\right.\)
=> \(\left[{}\begin{matrix}a=2\\a=-6\end{matrix}\right.\)
- Thay lại \(x^2+x=a\) vào phương trình trên ta được :\(\left[{}\begin{matrix}x^2+x=2\\x^2+x=-6\end{matrix}\right.\)
=> \(\left[{}\begin{matrix}x^2+x-2=0\\x^2+x+6=0\end{matrix}\right.\)
=> \(\left[{}\begin{matrix}\left(x+\frac{1}{2}\right)^2-\frac{9}{4}=0\\\left(x+\frac{1}{2}\right)^2+\frac{23}{4}=0\end{matrix}\right.\)
=> \(\left[{}\begin{matrix}\left(x+\frac{1}{2}\right)^2=\frac{9}{4}\\\left(x+\frac{1}{2}\right)^2=-\frac{23}{4}\left(VL\right)\end{matrix}\right.\)
=> \(\left[{}\begin{matrix}x+\frac{1}{2}=\sqrt{\frac{9}{4}}\\x+\frac{1}{2}=-\sqrt{\frac{9}{4}}\end{matrix}\right.\)
=> \(\left[{}\begin{matrix}x=\sqrt{\frac{9}{4}}-\frac{1}{2}=1\\x=-\sqrt{\frac{9}{4}}-\frac{1}{2}=-2\end{matrix}\right.\)
Vậy phương trình trên có nghiệm là \(S=\left\{1,-2\right\}\)
b, Đặt \(x^2+2x+3=a\) -> làm tương tự câu a .
c, Ta có : \(\left(x-2\right)\left(x+2\right)\left(x^2-10\right)=72\)
=> \(\left(x^2-4\right)\left(x^2-10\right)=72\)
- Đặt \(x^2-4=a\) và \(x^2-10=a-6\) ta được phương trình :
\(a\left(a-6\right)=72\)
=> \(a^2-6a-72=0\)
=> \(a^2+6a-12a-72=0\)
=> \(a\left(a+6\right)-12\left(a+6\right)=0\)
=> \(\left(a+6\right)\left(a-12\right)=0\)
=> \(\left[{}\begin{matrix}a+6=0\\a-12=0\end{matrix}\right.\)
=> \(\left[{}\begin{matrix}a=-6\\a=12\end{matrix}\right.\)
- Thay lại \(x^2-4=a\) vào phương trình trên ta được :\(\left[{}\begin{matrix}x^2-4=-6\\x^2-4=12\end{matrix}\right.\)
=> \(\left[{}\begin{matrix}x^2=-2\left(VL\right)\\x^2=16\end{matrix}\right.\)
=> \(\left[{}\begin{matrix}x=\sqrt{16}=4\\x=-\sqrt{16}=-4\end{matrix}\right.\)
Vậy phương trình trên có nghiệm là \(S=\left\{4,-4\right\}\)
d, Ta có : \(x\left(x+1\right)\left(x^2+x+1\right)=42\)
=> \(\left(x^2+x\right)\left(x^2+x+1\right)=42\)
- Đặt \(x^2+x=a\) ta được phương trình : \(a\left(a+1\right)=42\)
=> \(a^2+a-42=0\)
=> \(a^2+7a-6a-42=0\)
=> \(a\left(a+7\right)-6\left(a+7\right)=0\)
=> \(\left(a-6\right)\left(a+7\right)=0\)
=> \(\left[{}\begin{matrix}a=6\\a=-7\end{matrix}\right.\)
- Thay \(a=x^2+x\) vào phương trình ta được : \(\left[{}\begin{matrix}x^2+x=6\\x^2+x=-7\end{matrix}\right.\)
=> \(\left[{}\begin{matrix}x^2+x-6=0\\x^2+x+7=0\end{matrix}\right.\)
=> \(\left[{}\begin{matrix}\left(x+\frac{1}{2}\right)^2-\frac{25}{4}=0\\\left(x+\frac{1}{2}\right)^2+\frac{27}{4}=0\end{matrix}\right.\)
=> \(\left[{}\begin{matrix}\left(x+\frac{1}{2}\right)^2=\frac{25}{4}\\\left(x+\frac{1}{2}\right)^2=-\frac{27}{4}\left(VL\right)\end{matrix}\right.\)
=> \(\left[{}\begin{matrix}x+\frac{1}{2}=\sqrt{\frac{25}{4}}\\x+\frac{1}{2}=-\sqrt{\frac{25}{4}}\end{matrix}\right.\)
=> \(\left[{}\begin{matrix}x=\sqrt{\frac{25}{4}}-\frac{1}{2}=2\\x=-\sqrt{\frac{25}{4}}-\frac{1}{2}=-3\end{matrix}\right.\)
Vậy phương trình trên có tập nghiệm là \(S=\left\{2;-3\right\}\)
a, \(\left(x^2+x\right)^2+4\left(x^2+x\right)-12=0\)
\(\Leftrightarrow x^4+2x^3+x^2+4x^2+4x+12=0\)
\(\Leftrightarrow x^4+2x^3+5x^2+4x-12=0\)
\(\Leftrightarrow x^4-x^3+3x^3-3x^2+8x^2-8x+12x-12=0\)
\(\Leftrightarrow x^3\left(x-1\right)+3x^2\left(x-1\right)+8x\left(x-1\right)+12\left(x-1\right)=0\)
\(\Leftrightarrow\left(x^3+3x^2+8x+12\right)\left(x-1\right)=0\)
\(\Leftrightarrow\left(x^3+2x^2+x^2+2x+6x+12\right)\left(x-1\right)=0\)
\(\Leftrightarrow\left[x^2\left(x+2\right)+x\left(x+2\right)+6\left(x+2\right)\right]\left(x-1\right)=0\)
\(\Leftrightarrow\left(x^2+x+6\right)\left(x+2\right)\left(x-1\right)=0\)
có : \(x^2+x+6>0\)
\(\Leftrightarrow\orbr{\begin{cases}x+2=0\\x-1=0\end{cases}\Leftrightarrow\orbr{\begin{cases}x=-2\\x=1\end{cases}}}\)
b, \(\left(x-1\right)\left(x-3\right)\left(x+5\right)\left(x+7\right)-297=0\)
\(\Leftrightarrow\left[\left(x-1\right)\left(x+5\right)\right]\left[\left(x-3\right)\left(x+7\right)\right]-297=0\)
\(\Leftrightarrow\left(x^2+4x-5\right)\left(x^2+7x-21\right)-297=0\)
đặt \(x^2+4x-13=t\)
\(\Leftrightarrow\left(t+8\right)\left(t-8\right)-297=0\)
\(\Leftrightarrow t^2-64-297=0\)
\(\Leftrightarrow t^2=361\)
\(\Leftrightarrow t=\pm19\)
có t rồi tìm x thôi
Lời giải:
a)
\((x-2)(x-3)+2x=(x-2)^2-2\)
\(\Leftrightarrow (x-2)(x-2-1)+2x=(x-2)^2-2\)
\(\Leftrightarrow (x-2)^2-(x-2)+2x=(x-2)^2-2\)
\(\Leftrightarrow x+4=0\Rightarrow x=-4\)
b)
\((x-1)^2+3x(x-1)+7=(2x-1)^2+5(x-3)\)
\(\Leftrightarrow (x-1)^2+3x(x-1)+7=x^2+(x-1)^2+2x(x-1)+5(x-3)\)
\(\Leftrightarrow x(x-1)+7=x^2+5(x-3)\)
\(\Leftrightarrow 6x=22\Rightarrow x=\frac{11}{3}\)
c)
\(5(x^2-2x-1)+2(3x-2)=5(x+1)^2=5(x^2-2x+1)\)
\(\Leftrightarrow -5+2(3x-2)=5\)
\(\Leftrightarrow 3x-2=5\Rightarrow x=\frac{7}{3}\)
d)
\((x-1)(x^2+x+1)-2x=x(x-1)(x+1)=x(x^2-1)\)
\(\Leftrightarrow x^3-1-2x=x^3-x\Leftrightarrow -1-x=0\Rightarrow x=-1\)
a) \(\dfrac{3x-1}{x-1}-\dfrac{2x+5}{x+3}=1-\dfrac{4}{\left(x-1\right)\left(x+3\right)}\)
ĐKXĐ \(x-1\ne0\) hoặc \(x+3\ne0\)
\(\Rightarrow x\ne1\) và \(x\ne-3\)
\(\dfrac{\left(3x-1\right)\left(x+3\right)}{\left(x-1\right)\left(x+3\right)}-\dfrac{\left(2x+5\right)\left(x-1\right)}{\left(x-1\right)\left(x+3\right)}=\dfrac{\left(x-1\right)\left(x+3\right)}{\left(x-1\right)\left(x+3\right)}-\dfrac{4}{\left(x-1\right)\left(x+3\right)}\)
\(\Leftrightarrow\left(3x-1\right)\left(x+3\right)-\left(2x+5\right)\left(x-1\right)=\left(x-1\right)\left(x+3\right)-4\)
\(\Leftrightarrow3x^2+9x-x-3-\left(2x^2-2x+5x-5\right)=x^2+3x-x-3-4\)
\(\Leftrightarrow3x^2+9x-x-3-2x^2+2x-5x+5=x^2+3x-x-3-4\)
\(\Leftrightarrow9x-x+2x-5x-3x+x=3-5-3-4\)
\(\Leftrightarrow3x=-9\)
\(\Leftrightarrow x=-3\) (không thỏa ĐK)
Vậy PTVN
b) \(\dfrac{13}{\left(x-3\right)\left(2x+7\right)}+\dfrac{1}{2x+7}=\dfrac{6}{\left(x-3\right)\left(x+3\right)}\)
ĐKXĐ: \(x-3\ne0\Rightarrow x\ne3\)
\(x+3\ne0\Rightarrow x\ne-3\)
\(2x+7\ne0\Rightarrow2x\ne-7\Rightarrow x\ne\dfrac{-7}{2}\)
\(\dfrac{13\left(x+3\right)}{\left(x-3\right)\left(x+3\right)\left(2x+7\right)}+\dfrac{\left(x-3\right)\left(x+3\right)}{\left(x-3\right)\left(x+3\right)\left(2x+7\right)}=\dfrac{6\left(2x+7\right)}{\left(x-3\right)\left(x+3\right)\left(2x+7\right)}\)
\(\Leftrightarrow13\left(x+3\right)+\left(x-3\right)\left(x+3\right)=6\left(2x+7\right)\)
\(\Leftrightarrow13x+39+x^2+3x-3x-9=12x+42\)
\(\Leftrightarrow x^2+x-12=0\)
\(\Leftrightarrow x^2-3x+4x-12=0\)
\(\Leftrightarrow x\left(x-3\right)+4\left(x-3\right)=0\)
\(\Leftrightarrow\left(x-3\right)\left(x+4\right)=0\)
\(\left\{{}\begin{matrix}x-3=0\\x+4=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=3\left(KTĐK\right)\\x=-4\left(TĐK\right)\end{matrix}\right.\)
Vậy S={-4}
a) \(\dfrac{3x-1}{x-1}-\dfrac{2x+5}{x+3}=1-\dfrac{4}{\left(x-1\right)\left(x+3\right)}\) ( đk: x ≠ 1 ; x ≠ -3 )
\(\Leftrightarrow\left(3x-1\right)\left(x+3\right)-\left(2x+5\right)\left(x-1\right)=\left(x-1\right)\left(x+3\right)-4\)
\(\Leftrightarrow3x^2+9x-x-3-2x^2+2x-5x+5=x^2+3x-x-3-4\)
\(\Leftrightarrow3x=-9\)
\(\Rightarrow x=-3\left(KTM\right)\)
S = ∅
b) \(\dfrac{13}{\left(x-3\right)\left(2x+7\right)}+\dfrac{1}{2x+7}=\dfrac{6}{\left(x-3\right)\left(x+3\right)}\)
( đk: x ≠ ± 3 ; x ≠ \(\dfrac{-7}{2}\) )
\(\Leftrightarrow13\left(x+3\right)+\left(x-3\right)\left(x+3\right)=6\left(2x+7\right)\)
\(\Leftrightarrow13x+39+x^2-9=12x+42\)
\(\Leftrightarrow x^2-x-12=0\)
\(\Leftrightarrow x^2+3x-4x-12=0\)
\(\Leftrightarrow x\left(x+3\right)-4\left(x+3\right)=0\)
\(\Leftrightarrow\left(x-4\right)\left(x+3\right)=0\)
\(\Rightarrow\left\{{}\begin{matrix}x-4=0\\x-3=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x=4\left(TM\right)\\x=3\left(KTM\right)\end{matrix}\right.\)
S = \(\left\{4\right\}\)
( x - 1 ) ( x - 3 ) ( x + 5 ) ( x + 7) - 297 = 0
<=> ( x2 + 4x - 5 ) ( x2 + 4x - 21 ) - 297 = 0
Đặt x2 + 4x - 5 = t ( t > -9 )
Ta có : t (t - 16 ) - 297 = 0 <=> t2 - 16t - 297 = 0 <=> t = 27 ; t = 11 ( loại)
Ta có x2 + 4x - 5 = 27 <=> x2 + 4x - 32 = 0 <=> x = 4 , x = -8
a/ Đặt (x^2 - 5x) = a thì ta có
a^2 + 10a + 24 = 0
<=> (a + 4)(a + 6) = 0
Làm nốt
b/ (x - 4)(x - 5)(x - 6)(x - 7) = 1680
<=> (x - 4)(x - 7)(x - 5)(x - 6) = 1680
<=> (x^2 - 11x + 28)(x^2 - 11x + 30) = 1680
Đặt x^2 - 11x + 28 = a thì ta có
a(a + 2) = 1680
<=> (a - 40)(a + 42) = 0
Làm nốt
a); b) Do tích = 0
=> Từng thừa số = 0 và ta nhận xét: \(x^2+2;x^2+3>0\)
=> a) \(\orbr{\begin{cases}x=1\\x=-\frac{5}{2}\end{cases}}\)
và câu b) \(\orbr{\begin{cases}x=\frac{1}{2}\\x=5\end{cases}}\)
a; *x-1=0 <=>x=1
*2x+5=0 <=>x=-2,5
*x2+2=0 <=> ko có x
b; tương tự a
Bài 1:
a) Ta có: \(\frac{4}{5}x-3=\frac{1}{5}x\left(4x-15\right)\)
\(\Leftrightarrow\frac{4x}{5}-3=\frac{4x^2}{5}-3x\)
\(\Leftrightarrow\frac{12x}{15}-\frac{45}{15}-\frac{12x^2}{15}+\frac{45x}{15}=0\)
Suy ra: \(12x-45-12x^2+45x=0\)
\(\Leftrightarrow-12x^2+57x-45=0\)
\(\Leftrightarrow-12x^2+12x+45x-45=0\)
\(\Leftrightarrow-12x\left(x-1\right)+45\left(x-1\right)=0\)
\(\Leftrightarrow\left(x-1\right)\left(-12x+45\right)=0\)
\(\Leftrightarrow-3\left(x-1\right)\left(4x-15\right)=0\)
mà \(-3\ne0\)
nên \(\left[{}\begin{matrix}x-1=0\\4x-15=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=1\\4x=15\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=1\\x=\frac{15}{4}\end{matrix}\right.\)
Vậy: Tập nghiệm \(S=\left\{1;\frac{15}{4}\right\}\)
b) Ta có: \(\left(x-3\right)-\frac{\left(x-3\right)\left(2x-5\right)}{6}=\frac{\left(x-3\right)\left(3-x\right)}{4}\)
\(\Leftrightarrow\left(x-3\right)-\frac{\left(x-3\right)\left(2x-5\right)}{6}+\frac{\left(x-3\right)^2}{4}=0\)
\(\Leftrightarrow\frac{12\left(x-3\right)}{12}-\frac{2\left(x-3\right)\left(2x-5\right)}{12}+\frac{3\left(x-3\right)^2}{12}=0\)
Suy ra: \(12\left(x-3\right)-2\left(2x^2-11x+15\right)+3\left(x^2-6x+9\right)=0\)
\(\Leftrightarrow12x-36-4x^2+22x-30+3x^2-18x+27=0\)
\(\Leftrightarrow-x^2+16x-39=0\)
\(\Leftrightarrow-\left(x^2-16x+39\right)=0\)
\(\Leftrightarrow x^2-13x-3x+39=0\)
\(\Leftrightarrow x\left(x-13\right)-3\left(x-13\right)=0\)
\(\Leftrightarrow\left(x-13\right)\left(x-3\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x-13=0\\x-3=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=13\\x=3\end{matrix}\right.\)
Vậy: Tập nghiệm S={3;13}
c) Ta có: \(\frac{\left(3x+1\right)\left(3x-2\right)}{3}+5\left(3x+1\right)=\frac{2\left(2x+1\right)\left(3x+1\right)}{3}+2x\left(3x+1\right)\)
\(\Leftrightarrow\frac{9x^2-3x-2}{3}+5\left(3x+1\right)-\frac{12x^2+10x+2}{3}-2x\left(3x+1\right)=0\)
\(\Leftrightarrow\frac{9x^2-3x-2-12x^2-10x-2}{3}-6x^2+13x+5=0\)
\(\Leftrightarrow\frac{-3x^2-13x-4}{3}+\frac{3\left(-6x^2+13x+5\right)}{3}=0\)
Suy ra: \(-3x^2-13x-4-18x^2+39x+15=0\)
\(\Leftrightarrow-21x^2+26x+11=0\)
\(\Leftrightarrow-21x^2-7x+33x+11=0\)
\(\Leftrightarrow-7x\left(3x+1\right)+11\left(3x+1\right)=0\)
\(\Leftrightarrow\left(3x+1\right)\left(-7x+11\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}3x+1=0\\-7x+11=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}3x=-1\\-7x=-11\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=\frac{-1}{3}\\x=\frac{11}{7}\end{matrix}\right.\)
Vậy: Tập nghiệm \(S=\left\{-\frac{1}{3};\frac{11}{7}\right\}\)
\(\left(x^2+4x-5\right)\left(x^2+4x-21\right)=297\)
đặt a = \(x^2+4x-5\) vào bt ta được:
\(a\left(a-16\right)-297=0\Leftrightarrow a^2-16a+64-361=0\)
\(\Leftrightarrow\left(a-8\right)^2-19^2=0\Leftrightarrow\left(a-27\right)\left(a+11\right)=0\)
\(\Leftrightarrow\left(x^2+4x-32\right)\left(x^2+4+6\right)=0\)
\(\Leftrightarrow\left(x-4\right)\left(x+8\right)\left(\left(x+2\right)^2+2\right)=0\)
\(\left\{{}\begin{matrix}x-4=0\\x+8=0\\\left(x+2\right)^2=-2\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x=4\\x=-8\end{matrix}\right.\)
\(\left(x-1\right)\left(x-3\right)\left(x+5\right)\left(x+7\right)=297\)
\(\Leftrightarrow\left(x^2-3x-x+3\right)\left(x^2+7x+5x+35\right)=297\)
\(\Leftrightarrow\left(x^2-4x+3\right)\left(x^2+12x+35\right)=297\)
\(\Leftrightarrow x^4+12x^3+35x^2-4x^3-48x^2-140x+3x^2+36x+105=297\)
\(\Leftrightarrow x^4+8x^3-10x^2-104x+105-297=0\)
\(\Leftrightarrow x^4-4x^3+12x^3-48x^2+38x^2-152x+48x-192=0\)
\(\Leftrightarrow x^3\left(x-4\right)+12x^2\left(x-4\right)+38x\left(x-4\right)+48\left(x-4\right)=0\)
\(\Leftrightarrow\left(x-4\right)\left(x^3+12x^2+38x+48\right)=0\)
\(\Leftrightarrow\left(x-4\right)\left(x^3+8x^2+4x^2+32x+6x+48\right)=0\)
\(\Leftrightarrow\left(x-4\right)\left[x^2\left(x+8\right)+4x\left(x+8\right)+6\left(x+8\right)\right]=0\)
\(\Leftrightarrow\left(x-4\right)\left(x+8\right)\left(x^2+4x+6\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x-4=0\\x+8=0\\x^2+4x+6=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=4\\x=-8\\x\notin R\end{matrix}\right.\)
\(S=\left\{4;-8\right\}\)