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5/
Đặt \(\left\{{}\begin{matrix}\sqrt{2x-\frac{3}{x}}=a\ge0\\\sqrt{\frac{6}{x}-2x}=b\ge0\end{matrix}\right.\) \(\Rightarrow a^2+b^2=\frac{3}{x}\)
Pt trở thành:
\(a-1=\frac{a^2+b^2}{2}-b\)
\(\Leftrightarrow a^2+b^2-2a-2b+2=0\)
\(\Leftrightarrow\left(a^2-2a+1\right)+\left(b^2-2b+1\right)=0\)
\(\Leftrightarrow\left(a-1\right)^2+\left(b-1\right)^2=0\)
\(\Leftrightarrow\left\{{}\begin{matrix}a=1\\b=1\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\sqrt{2x-\frac{3}{x}}=1\\\sqrt{\frac{6}{x}-2x}=1\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}2x^2-x-3=0\\2x^2+x-6=0\end{matrix}\right.\) \(\Rightarrow x=\frac{3}{2}\)
4/
ĐKXĐ: \(x\ge\frac{1}{5}\)
\(\Leftrightarrow\frac{4x-3}{\sqrt{5x-1}+\sqrt{x+2}}=\frac{4x-3}{5}\)
\(\Leftrightarrow\left[{}\begin{matrix}4x-3=0\Rightarrow x=\frac{3}{4}\\\sqrt{5x-1}+\sqrt{x+2}=5\left(1\right)\end{matrix}\right.\)
\(\left(1\right)\Leftrightarrow\sqrt{5x-1}-3+\sqrt{x+2}-2=0\)
\(\Leftrightarrow\frac{5\left(x-2\right)}{\sqrt{5x-1}+3}+\frac{x-2}{\sqrt{x+2}+2}=0\)
\(\Leftrightarrow\left(x-2\right)\left(\frac{5}{\sqrt{5x-1}+3}+\frac{1}{\sqrt{x+2}+2}\right)=0\)
\(\Leftrightarrow x=2\)
mấy bài này thì bạn cứ đặt ẩn phụ cho dễ nhìn hơn mà giải nhé
a, \(\hept{\begin{cases}\frac{1}{2x-y}+x+3y=\frac{3}{2}\\\frac{4}{2x-y}-5\left(x+3y\right)=-3\end{cases}}\)ĐK : \(2x\ne y\)
Đặt \(\frac{1}{2x-y}=t;x+3y=u\)hệ phương trình tương đương
\(\hept{\begin{cases}t+u=\frac{3}{2}\\4t-5u=-3\end{cases}\Leftrightarrow\hept{\begin{cases}4t+4u=6\\4t-5u=-3\end{cases}}\Leftrightarrow\hept{\begin{cases}9u=9\\4t=-3+5u\end{cases}}\Leftrightarrow\hept{\begin{cases}u=1\\t=\frac{-3+5}{4}=\frac{1}{2}\end{cases}}}\)
Theo cách đặt \(\hept{\begin{cases}x+3y=1\\\frac{1}{2x-y}=\frac{1}{2}\end{cases}\Leftrightarrow\hept{\begin{cases}x+3y=1\\2x-y=2\end{cases}}\Leftrightarrow\hept{\begin{cases}2x+6y=2\\2x-y=2\end{cases}\Leftrightarrow}\hept{\begin{cases}7y=4\\x=\frac{y+2}{2}\end{cases}\Leftrightarrow}\hept{\begin{cases}y=\frac{4}{7}\\x=\frac{9}{7}\end{cases}}}\)
Vậy hệ pt có một nghiệm (x;y) = (9/7;4/7)
A = \(x^2+3x-7=x^2+2x\frac{3}{2}+\frac{9}{4}-\frac{37}{4}\)
\(=\left(x+\frac{3}{2}\right)^2-\frac{37}{4}\ge-\frac{37}{4}\)
\(\Rightarrow\)min A = \(-\frac{37}{4}\Leftrightarrow x=-\frac{3}{2}\)
B = \(x-5\sqrt{x}-1\) ĐKXĐ: \(x\ge0\)
\(=x-2\sqrt{x}\frac{5}{2}+\frac{25}{4}-\frac{29}{4}=\left(\sqrt{x}-\frac{5}{2}\right)^2-\frac{29}{4}\ge-\frac{29}{4}\)
\(\Rightarrow\)min B = \(-\frac{29}{4}\Leftrightarrow x=\frac{25}{4}\)( thỏa mãn)
C = \(\frac{-4}{\sqrt{x}+7}\) ĐKXĐ:\(x\ge0\)
Ta có: \(\sqrt{x}+7\ge7\Rightarrow\frac{4}{\sqrt{x}+7}\le\frac{4}{7}\)\(\Leftrightarrow\frac{-4}{\sqrt{x}+7}\ge-\frac{4}{7}\)
\(\Rightarrow\)min C = \(-\frac{4}{7}\Leftrightarrow x=0\)
D = \(\frac{\sqrt{x}+1}{\sqrt{x}+3}\) ĐKXĐ:\(x\ge0\)
\(=1-\frac{2}{\sqrt{x}+3}\ge1-\frac{2}{3}=\frac{1}{3}\)
\(\Rightarrow\)min D = \(\frac{1}{3}\Leftrightarrow x=0\)
E = \(\frac{x+7}{\sqrt{x}+3}\) ĐKXĐ:\(x\ge0\)
\(=\frac{x-9+16}{\sqrt{x}+3}=\frac{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)+16}{\sqrt{x}+3}=\sqrt{x}-3+\frac{16}{\sqrt{x}+3}=\sqrt{x}+3+\frac{16}{\sqrt{x}+3}-6\ge2\sqrt{16}-6=2\)
\(\Rightarrow\)min E = \(2\Leftrightarrow x=1\)(thỏa mãn)
F = \(\frac{x^2+3x+5}{x^2}\) ĐKXĐ: \(x\ne0\)
\(\Leftrightarrow\)\(x^2\left(F-1\right)-3x-5=0\)
△ = \(3^2+20\left(F-1\right)\ge0\)\(\Leftrightarrow F\ge\frac{11}{20}\)
\(\Rightarrow\)min F = \(\frac{11}{20}\Leftrightarrow x=-\frac{10}{3}\)( thỏa mãn)
giải pt
\(|4x-1|\)\(\sqrt{x^2+1}\)=2\(x^2\) -2x+2
\(\sqrt{\frac{1}{x+3}}\)+\(\sqrt{\frac{5}{x+4}}\) =4
a,\(\Leftrightarrow\left(4x-1\right)^2\left(x^2+1\right)=4\left(x^2-x+1\right)^2\)
\(\Leftrightarrow\left(16x^2-8x+1\right)\left(x^2+1\right)=4\left(x^4+x^2+1-2x^3+2x^2-2x\right)\)
\(\Leftrightarrow16x^4+17x^2-8x^3-8x+1=4x^4+12x^2+4-8x^3-8x\)
\(\Leftrightarrow12x^4+5x^2-3=0\left(1\right)\)
Dat \(x^2=t\left(t\ge0\right)\)
\(\left(1\right)\Leftrightarrow12t^2+5t-3=0\)
\(\Delta=25-4.12.\left(-3\right)=169>0\)
Suy ra PT co hai nghiem phan biet
\(t_1=\frac{1}{3};t_2=-\frac{3}{4}\)
\(x=\frac{1}{\sqrt{3}}\)
6/ Đặt \(\hept{\begin{cases}\sqrt[4]{x}=a\\\sqrt[4]{2-x}=b\end{cases}}\)
\(\Rightarrow b^4+a^4=2\)
Từ đó ta có: a + b = 2
Ta có: \(a^4+b^2\ge\frac{\left(a^2+b^2\right)^2}{2}\ge\frac{\left(a+b\right)^4}{8}=\frac{16}{8}=2\)
Dấu = xảy ra khi a = b = 1
=> x = 1