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Bài 4:
a) Thay x=49 vào B ta có:
\(B=\dfrac{1-\sqrt{49}}{1+\sqrt{49}}=-\dfrac{3}{4}\)
b) \(A=\left(\dfrac{15-\sqrt{x}}{x-25}+\dfrac{2}{\sqrt{x}+5}\right):\dfrac{\sqrt{x}+1}{\sqrt{x}-5}\)
\(A=\left[\dfrac{15-\sqrt{x}}{\left(\sqrt{x}+5\right)\left(\sqrt{x}-5\right)}+\dfrac{2\left(\sqrt{x}-5\right)}{\left(\sqrt{x}+5\right)\left(\sqrt{x}-5\right)}\right]\cdot\dfrac{\sqrt{x}-5}{\sqrt{x}+1}\)
\(A=\dfrac{15-\sqrt{x}+2\sqrt{x}-10}{\left(\sqrt{x}+5\right)\left(\sqrt{x}-5\right)}\cdot\dfrac{\sqrt{x}-5}{\sqrt{x}+1}\)
\(A=\dfrac{\sqrt{x}+5}{\left(\sqrt{x}+5\right)\left(\sqrt{x}-5\right)}\cdot\dfrac{\sqrt{x}-5}{\sqrt{x}+1}\)
\(A=\dfrac{1}{\sqrt{x}-5}\cdot\dfrac{\sqrt{x}-5}{\sqrt{x}+1}\)
\(A=\dfrac{1}{\sqrt{x}+1}\)
c) Ta có:
\(M=A-B=\dfrac{1}{\sqrt{x}+1}-\dfrac{1-\sqrt{x}}{\sqrt{x}+1}\)
\(M=\dfrac{1-1+\sqrt{x}}{\sqrt{x}+1}\)
\(M=\dfrac{\sqrt{x}}{\sqrt{x}+1}\)
\(M=\dfrac{\sqrt{x}+1-1}{\sqrt{x}+1}=\dfrac{\sqrt{x}+1}{\sqrt{x}+1}-\dfrac{1}{\sqrt{x}+1}=1-\dfrac{1}{\sqrt{x}+1}\)
Mà M nguyên khi:
\(1\) ⋮ \(\sqrt{x}+1\)
\(\Rightarrow\sqrt{x}+1\in\left\{1;-1\right\}\)
Mà: \(\sqrt{x}+1\ge1\)
\(\Rightarrow\sqrt{x}+1=1\)
\(\Rightarrow\sqrt{x}=0\)
\(\Rightarrow x=0\left(tm\right)\)
Vậy M nguyên khi x=0
Ta có: \(\Delta=7^2-4\cdot2\cdot\left(-1\right)=49+9=58>0\)
nên phương trình có hai nghiệm phân biệt
a) Áp dụng hệ thức Vi-et, ta có:
\(\left\{{}\begin{matrix}x_1\cdot x_2=\dfrac{-1}{2}\\x_1+x_2=\dfrac{-7}{2}\end{matrix}\right.\)
Ta có: \(\dfrac{1}{x_1^2}+\dfrac{1}{x_2^2}\)
\(=\dfrac{x_1^2+x_2^2}{\left(x_1\cdot x_2\right)^2}\)
\(=\dfrac{\left(x_1+x_2\right)^2-2\cdot x_1\cdot x_2}{\left(x_1\cdot x_2\right)^2}\)
\(=\dfrac{\left(-\dfrac{7}{2}\right)^2-2\cdot\dfrac{-1}{2}}{\left(-\dfrac{1}{2}\right)^2}=\dfrac{\dfrac{49}{4}+1}{\dfrac{1}{4}}=\dfrac{53}{4}\cdot4=53\)
\(\left\{{}\begin{matrix}x_1+x_2=-\dfrac{7}{2}\\x_1x_2=-\dfrac{1}{2}\end{matrix}\right.\)
\(\dfrac{1}{x_1^3}+\dfrac{1}{x_2^3}=\dfrac{x_1^3+x_2^3}{\left(x_1x_2\right)^3}=\dfrac{\left(x_1+x_2\right)^3-3x_1x_2\left(x_1+x_2\right)}{\left(x_1x_2\right)^3}=\dfrac{\left(-\dfrac{7}{2}\right)^3-3.\left(-\dfrac{1}{2}\right).\left(-\dfrac{7}{2}\right)}{\left(-\dfrac{1}{2}\right)^3}=385\)
\(x_1^3+x_2^3=\left(x_1+x_2\right)^3-3x_1x_2\left(x_1+x_2\right)=\left(-\dfrac{7}{2}\right)^3-3\left(-\dfrac{1}{2}\right).\left(-\dfrac{7}{2}\right)=-\dfrac{385}{8}\)
\(n=\sqrt{2}\left(\sqrt{3}+1\right)\sqrt{2-\sqrt{3}}\\ n=\left(\sqrt{3}+1\right)\sqrt{4-2\sqrt{3}}\\ n=\left(\sqrt{3}+1\right)\sqrt{\left(\sqrt{3}-1\right)^2}\\ n=\left(\sqrt{3}+1\right)\left|\sqrt{3}-1\right|\\ n=\left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right)\\ n=3-1=2\)
\(b,\dfrac{\sqrt{12}-\sqrt{6}}{\sqrt{30}-\sqrt{15}}=\dfrac{\sqrt{6}\left(\sqrt{2}-1\right)}{\sqrt{15}\left(\sqrt{2}-1\right)}=\dfrac{\sqrt{6}}{\sqrt{15}}=\dfrac{\sqrt{2}}{\sqrt{5}}\)
\(d,\dfrac{ab-bc}{\sqrt{ab}-\sqrt{bc}}=\dfrac{\left(\sqrt{ab}-\sqrt{bc}\right)\left(\sqrt{ab}+\sqrt{bc}\right)}{\left(\sqrt{ab}-\sqrt{bc}\right)}=\sqrt{ab}+\sqrt{bc}=\sqrt{b}\left(\sqrt{a}+\sqrt{c}\right)\)
\(e,\left(a\sqrt{\dfrac{a}{b}+2\sqrt{ab}}+b\sqrt{\dfrac{a}{b}}\right)\sqrt{ab}\)
\(=a\left(\sqrt{\dfrac{a}{b}+\dfrac{2b.\sqrt{ab}}{b}}+b\sqrt{\dfrac{a}{b}}\right)\sqrt{ab}\)
\(=a\sqrt{a}\sqrt{a+2b\sqrt{ab}}+b\sqrt{a^2}\)
\(=a\sqrt{a^2+2ab\sqrt{ab}}+ab\)
\(=a\left(\sqrt{a^2+2ab\sqrt{ab}}+b\right)\)
\(f,\left(\dfrac{1-a\sqrt{a}}{1-\sqrt{a}}+\sqrt{a}\right)\left(\dfrac{1+a\sqrt{a}}{1+\sqrt{a}}-\sqrt{a}\right)\)
\(=\left(a+\sqrt{a}+1+\sqrt{a}\right)\left(a-\sqrt{a}+1-\sqrt{a}\right)\)
\(=\left(a+2\sqrt{a}+1\right)\left(a-2\sqrt{a}+1\right)\)
\(=\left(\sqrt{a}+1\right)^2\left(\sqrt{a}-1\right)^2\)
\(=\left(a-1\right)^2=a^2-2a+1\)
\(b,\Leftrightarrow\left\{{}\begin{matrix}a=2;b\ne3\\2a+b=1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}a=2\\b=-3\end{matrix}\right.\\ b,\text{Gọi }M\left(x_0;y_0\right)\text{ là điểm cần tìm }\Leftrightarrow y_0=3x_0\\ M\left(x_0;y_0\right)\in\left(d\right)\Leftrightarrow2x_0+3=y_0=3x_0\Leftrightarrow x_0=3\Leftrightarrow y_0=9\\ \text{Vậy }M\left(3;9\right)\text{ là điểm cần tìm}\)