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\(x=\sqrt[3]{5-\sqrt{17}}+\sqrt[3]{5+\sqrt{17}}\)
\(\Leftrightarrow x^3=5-\sqrt{17}+5+\sqrt{17}+3\left(\sqrt[3]{5-\sqrt{17}}+\sqrt[3]{5+\sqrt{17}}\right)\sqrt[3]{5-\sqrt{17}}\sqrt[3]{5+\sqrt{17}}\)
\(\Leftrightarrow x^3=10+3x\sqrt[3]{\left(5-\sqrt{17}\right)\left(5+\sqrt{17}\right)}\)
\(\Leftrightarrow x^3=10+3x\sqrt[3]{8}\)\(\Leftrightarrow x^3=10+6x\)
\(\Leftrightarrow x^3-6x-10=0\)
Hay ta co DPCM
\(x^3=76+3\sqrt[3]{\left(38-17\sqrt{5}\right)\left(38+17\sqrt{5}\right)}\left(\sqrt[3]{38-17\sqrt{5}}+\sqrt[3]{38+17\sqrt{5}}\right)\)
\(\Leftrightarrow x^3=76-3x\)
\(\Leftrightarrow x^3+3x-76=0\)
\(\Leftrightarrow\left(x-4\right)\left(x^2+4x+19\right)=0\)
\(\Leftrightarrow x=4\)
\(\Rightarrow x^3-3x^2-2x-8=0\)
Đáp án :
\(x_0=^3\sqrt{38-17}\sqrt{5}+^3\sqrt{38+17}.\sqrt{5}\)
\(=x_0=38-17\sqrt{5}+38+17\sqrt{5}-3^3\sqrt{\left(38-17\sqrt{5}\right)\left(38+17\sqrt{5}\right).x_0}\)
\(=76-3^3\sqrt{-1}.x_0=76+3x_0\)
\(=x_0^3\)\(-3x_0-76=0\)
\(=\left(x_0-4\right)\left(x_0^2+4x_0+19\right)=0\)
\(=x_0=4\)
Thay x0 = 4 vào phương trình x3 - 3x2 - 2x - 8 = 0 ta có đẳng thức đúng là:
43 - 3.42 - 2.4 - 8 = 0
Vậy x0 là nghiệm của phương trình x3 - 3x2 - 2x - 8 = 0
\(x=\sqrt[3]{5-\sqrt{17}}+\sqrt[3]{5+\sqrt{17}}\)
\(\Rightarrow x^3=5-\sqrt{17}+5+\sqrt{17}+3\sqrt[3]{\left(5-\sqrt{17}\right)\left(5+\sqrt{17}\right)}x\)
\(\Rightarrow x^3=10+3\sqrt[3]{25-17}x\)
\(\Rightarrow x^3=10+3\sqrt[3]{8}x\)
\(\Rightarrow x^3=10+3.2x\)
\(\Leftrightarrow x^3-6x-10=0\)
Học toán vui vẻ!
a: Ta có: \(\sqrt{4x+20}-3\sqrt{x+5}+\dfrac{4}{3}\sqrt{9x+45}=6\)
\(\Leftrightarrow2\sqrt{x+5}-3\sqrt{x+5}+4\sqrt{x+5}=6\)
\(\Leftrightarrow3\sqrt{x+5}=6\)
\(\Leftrightarrow x+5=4\)
hay x=-1
b: Ta có: \(\dfrac{1}{2}\sqrt{x-1}-\dfrac{3}{2}\sqrt{9x-9}+24\sqrt{\dfrac{x-1}{64}}=-17\)
\(\Leftrightarrow\dfrac{1}{2}\sqrt{x-1}-\dfrac{9}{2}\sqrt{x-1}+3\sqrt{x-1}=-17\)
\(\Leftrightarrow\sqrt{x-1}=17\)
\(\Leftrightarrow x-1=289\)
hay x=290
a: ĐKXĐ: x>=-3/2
\(\sqrt{x^2+4}=\sqrt{2x+3}\)
=>\(x^2+4=2x+3\)
=>\(x^2-2x+1=0\)
=>\(\left(x-1\right)^2=0\)
=>x-1=0
=>x=1(nhận)
b: \(\sqrt{x^2-6x+9}=2x-1\)(ĐKXĐ: \(x\in R\))
=>\(\sqrt{\left(x-3\right)^2}=2x-1\)
=>\(\left\{{}\begin{matrix}\left(2x-1\right)^2=\left(x-3\right)^2\\x>=\dfrac{1}{2}\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}\left(2x-1-x+3\right)\left(2x-1+x-3\right)=0\\x>=\dfrac{1}{2}\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}\left(x+2\right)\left(3x-4\right)=0\\x>=\dfrac{1}{2}\end{matrix}\right.\)
=>x=4/3(nhận) hoặc x=-2(loại)
c:
Sửa đề: \(\sqrt{4x+12}=\sqrt{9x+27}-5\)
ĐKXĐ: \(x>=-3\)
\(\sqrt{4x+12}=\sqrt{9x+27}-5\)
=>\(2\sqrt{x+3}=3\sqrt{x+3}-5\)
=>\(-\sqrt{x+3}=-5\)
=>x+3=25
=>x=22(nhận)
d: ĐKXĐ: \(\left[{}\begin{matrix}x< =\dfrac{3-\sqrt{5}}{4}\\x>=\dfrac{3+\sqrt{5}}{4}\end{matrix}\right.\)
\(\sqrt{4x^2-6x+1}=\left|2x-5\right|\)
=>\(\sqrt{\left(4x^2-6x+1\right)}=\sqrt{4x^2-20x+25}\)
=>\(4x^2-6x+1=4x^2-20x+25\)
=>\(-6x+20x=25-1\)
=>\(14x=24\)
=>x=12/7(nhận)
\(x^3=10+3x\sqrt[3]{\left(5-\sqrt{17}\right)\left(5+\sqrt{17}\right)}=10+6x\)
Thay vào -> dpcm
\(x=\sqrt[3]{5-\sqrt{17}}+\sqrt[3]{5+\sqrt{17}}\)
\(\Leftrightarrow x^3=5-\sqrt{17}+5+\sqrt{17}\)
\(+3\left(\sqrt[3]{5-\sqrt{17}}+\sqrt[3]{5+\sqrt{17}}\right)\sqrt[3]{5-\sqrt{17}}\sqrt[3]{5+\sqrt{17}}\)
\(\Leftrightarrow x^3=10+3x\sqrt[3]{\left(5-\sqrt{17}\right)\left(5+\sqrt{17}\right)}\)
\(\Leftrightarrow x^3=10+3x\sqrt[3]{8}\Leftrightarrow x^3=10+6x\)
\(\Leftrightarrow x^3-6x-10=0\)
\(\Rightarrow\) Đpcm
Chúc bạn học tốt !!!