Giải phương trình:
a, \(x^2-3x+1=-\frac{\sqrt{3}}{3}\sqrt{x^4+x^2+1}\)
b, \(\left(x+3\right)\sqrt{\left(4-x\right)\left(12+x\right)}=28-x\)
c, \(\sqrt{x^3-x}=2x^2-x-2\)
d, \(2x^2+5+2\sqrt{x^2+x-2}=5\sqrt{x-1}+5\sqrt{x+2}\)
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c.
\(\Leftrightarrow x^2+3-\left(3x+1\right)\sqrt{x^2+3}+2x^2+2x=0\)
Đặt \(\sqrt{x^2+3}=t>0\)
\(\Rightarrow t^2-\left(3x+1\right)t+2x^2+2x=0\)
\(\Delta=\left(3x+1\right)^2-4\left(2x^2+2x\right)=\left(x-1\right)^2\)
\(\Rightarrow\left[{}\begin{matrix}t=\dfrac{3x+1-x+1}{2}=x+1\\t=\dfrac{3x+1+x-1}{2}=2x\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\sqrt{x^2+3}=x+1\left(x\ge-1\right)\\\sqrt{x^2+3}=2x\left(x\ge0\right)\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x^2+3=x^2+2x+1\left(x\ge-1\right)\\x^2+3=4x^2\left(x\ge0\right)\end{matrix}\right.\)
\(\Leftrightarrow x=1\)
a.
Đề bài ko chính xác, pt này ko giải được
b.
ĐKXĐ: \(x\ge-\dfrac{7}{2}\)
\(2x+7-\left(2x+7\right)\sqrt{2x+7}+x^2+7x=0\)
Đặt \(\sqrt{2x+7}=t\ge0\)
\(\Rightarrow t^2-\left(2x+7\right)t+x^2+7x=0\)
\(\Delta=\left(2x+7\right)^2-4\left(x^2+7x\right)=49\)
\(\Rightarrow\left[{}\begin{matrix}t=\dfrac{2x+7-7}{2}=x\\t=\dfrac{2x+7+7}{2}=x+7\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\sqrt{2x+7}=x\left(x\ge0\right)\\\sqrt{2x+7}=x+7\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x^2-2x-7=0\left(x\ge0\right)\\x^2+12x+42=0\left(vn\right)\end{matrix}\right.\)
\(\Rightarrow x=1+2\sqrt{2}\)
8.
ĐKXĐ: \(x\ge\frac{2}{3}\)
\(\Leftrightarrow\frac{9\left(x+3\right)}{\sqrt{4x+1}+\sqrt{3x-2}}=x+3\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-3\left(l\right)\\\frac{9}{\sqrt{4x+1}+\sqrt{3x-2}}=1\left(1\right)\end{matrix}\right.\)
\(\left(1\right)\Leftrightarrow\sqrt{4x+1}+\sqrt{3x-2}=9\)
\(\Leftrightarrow\sqrt{4x+1}-5+\sqrt{3x-2}-4=0\)
\(\Leftrightarrow\frac{4\left(x-6\right)}{\sqrt{4x+1}+5}+\frac{3\left(x-6\right)}{\sqrt{3x-2}+4}=0\)
\(\Leftrightarrow\left(x-6\right)\left(\frac{4}{\sqrt{4x+1}+5}+\frac{3}{\sqrt{3x-2}+4}\right)=0\)
\(\Leftrightarrow x=6\)
6.
ĐKXD: ...
\(\Leftrightarrow2\left(x^2-6x+9\right)+\left(x+5-4\sqrt{x+1}\right)=0\)
\(\Leftrightarrow2\left(x-3\right)^2+\frac{\left(x-3\right)^2}{x+5+4\sqrt{x+1}}=0\)
\(\Leftrightarrow\left(x-3\right)^2\left(2+\frac{1}{x+5+4\sqrt{x+1}}\right)=0\)
\(\Leftrightarrow x=3\)
7.
\(\sqrt{x-\frac{1}{x}}-\sqrt{2x-\frac{5}{x}}+\frac{4}{x}-x=0\)
Đặt \(\left\{{}\begin{matrix}\sqrt{x-\frac{1}{x}}=a\ge0\\\sqrt{2x-\frac{5}{x}}=b\ge0\end{matrix}\right.\) \(\Rightarrow a^2-b^2=\frac{4}{x}-x\)
\(\Rightarrow a-b+a^2-b^2=0\)
\(\Leftrightarrow\left(a-b\right)\left(a+b+1\right)=0\)
\(\Leftrightarrow a=b\Leftrightarrow x-\frac{1}{x}=2x-\frac{5}{x}\)
\(\Leftrightarrow x=\frac{4}{x}\Rightarrow x=\pm2\)
Thế nghiệm lại pt ban đầu để thử (hoặc là bạn tìm ĐKXĐ từ đầu)
a/
\(\Leftrightarrow2\left(x^2-x+1\right)-\left(x^2+x+1\right)=-\frac{\sqrt{3}}{3}\sqrt{\left(x^2-x+1\right)\left(x^2+x+1\right)}\)
Đặt \(\left\{{}\begin{matrix}\sqrt{x^2-x+1}=a>0\\\sqrt{x^2+x+1}=b>0\end{matrix}\right.\)
\(\Leftrightarrow6a^2+\sqrt{3}ab-3b^2=0\)
\(\Leftrightarrow\left(3a-\sqrt{3}b\right)\left(2a+\sqrt{3}b\right)=0\)
\(\Leftrightarrow3a-\sqrt{3}b=0\Rightarrow b=\sqrt{3}a\)
\(\Leftrightarrow\sqrt{x^2+x+1}=\sqrt{3}\sqrt{x^2-x+1}\)
\(\Leftrightarrow x^2+x+1=3x^2-3x+3\)
b/ ĐKXĐ: ...
Đặt \(\left\{{}\begin{matrix}x+3=a\\\sqrt{\left(4-x\right)\left(12+x\right)}=b\end{matrix}\right.\)
\(\Rightarrow a^2+b^2=x^2+6x+9+48-8x-x^2=57-2x=2\left(28-x\right)+1\)
\(\Rightarrow28-x=\frac{a^2+b^2-1}{2}\)
Phương trình trở thành:
\(ab=\frac{a^2+b^2-1}{2}\Leftrightarrow\left(a-b\right)^2=1\Leftrightarrow\left[{}\begin{matrix}a+1=b\\a-1=b\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x+4=\sqrt{\left(4-x\right)\left(12+x\right)}\\x+2=\sqrt{\left(4-x\right)\left(12+x\right)}\end{matrix}\right.\) \(\Leftrightarrow...\)
c/ ĐKXĐ: ...
\(\sqrt{x\left(x^2-1\right)}=2\left(x^2-1\right)-x\)
Đặt \(\left\{{}\begin{matrix}\sqrt{x}=a\ge0\\\sqrt{x^2-1}=b\ge0\end{matrix}\right.\)
\(ab=2a^2-b^2\Leftrightarrow2a^2-ab-b^2=0\)
\(\Leftrightarrow\left(a-b\right)\left(2a+b\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}a-b=0\\2a+b=0\left(l\right)\end{matrix}\right.\)
\(\Leftrightarrow a=b\Leftrightarrow\sqrt{x}=\sqrt{x^2-1}\)
\(\Leftrightarrow x^2-x-1=0\)
d/ Là \(2x^2+5\) hay \(2x+5\) bạn?