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Ta có : \(a-b=\sqrt{29+12\sqrt{5}}-2\sqrt{5}\)
\(=\sqrt{20+12\sqrt{5}+9}-2\sqrt{5}\)
\(=\sqrt{\left(2\sqrt{5}+3\right)^2}-2\sqrt{5}\)
\(=2\sqrt{5}+3-2\sqrt{5}\)
\(=3\).
\(\Rightarrow a=b+3\)
Thế vào A ta được :
\(A=\left(b+3\right)^2\left(b+4\right)-b^2\left(b-1\right)-11\left(b+3\right)b+2017\)
\(=b^3+10b^2+33b+36-b^3+b^2-11b^2-33b+2017\)
\(=2053\)
\(a-b=\sqrt{29+12\sqrt{5}}-2\sqrt{5}=\sqrt{9+2.3.2\sqrt{5}+20}-2\sqrt{5}=\sqrt{3^2+2.3.2\sqrt{5}+\left(2\sqrt{5}\right)^2}-2\sqrt{5}=\sqrt{\left(3+2\sqrt{5}\right)^2}-2\sqrt{5}=3+2\sqrt{5}-2\sqrt{5}=3\Leftrightarrow a=b+3\)
A=\(a^2\left(a+1\right)-b^2\left(b-1\right)-11ab+2017=\left(b+3\right)^2\left(b+3+1\right)-b^2\left(b-1\right)-11\left(b+3\right)b+2017=\left(b^2+6b+9\right)\left(b+4\right)-b^3+b^2-11b^2-33b+2017=b^3+4b^2+6b^2+24b+9b+36-b^3+b^2-11b^2-33b+2017=b^3+10b^2+9b+33b-b^3-10b^2-33b+2053=2053\Leftrightarrow A=2053\)
a) \(A=\left|2-\sqrt{5}\right|+\left|2\sqrt{2}-\sqrt{5}\right|\)
\(=\sqrt{5}-2+2\sqrt{2}-\sqrt{5}=2\sqrt{2}-2\)
b) \(B=\left|\sqrt{7}-2\sqrt{2}\right|+\left|3-2\sqrt{2}\right|\)
\(=2\sqrt{2}-7+3-2\sqrt{2}=-4\)
c) \(C=\sqrt{9+6\sqrt{2}+2}-\sqrt{9-6\sqrt{2}+2}\)
\(=\sqrt{\left(\sqrt{2}+3\right)^2}-\sqrt{\left(3-\sqrt{2}\right)^2}=\left(3+\sqrt{2}\right)-\left(3-\sqrt{2}\right)=2\sqrt{2}\)
d) \(D=\sqrt{9+12\sqrt{2}+8}+\sqrt{9-12\sqrt{2}+8}\)
\(=\sqrt{\left(3+2\sqrt{2}\right)^2}+\sqrt{\left(3-2\sqrt{2}\right)^2}=\left(3+2\sqrt{2}\right)-\left(3-2\sqrt{2}\right)=4\sqrt{2}\)
ai nay dung kinh nghiem la chinh
cau a)
ta thay \(10+6\sqrt{3}=\left(1+\sqrt{3}\right)^3\)
\(6+2\sqrt{5}=\left(1+\sqrt{5}\right)^2\)
khi do \(x=\frac{\sqrt[3]{\left(\sqrt{3}+1\right)^3}\left(\sqrt{3}-1\right)}{\sqrt{\left(1+\sqrt{5}\right)^2}-\sqrt{5}}\)
\(x=\frac{\left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right)}{1+\sqrt{5}-\sqrt{5}}\)
\(x=\frac{3-1}{1}=2\)
suy ra
x^3-4x+1=1
A=1^2018
A=1
b)
ta thay
\(7+5\sqrt{2}=\left(1+\sqrt{2}\right)^3\)
khi do
\(x=\sqrt[3]{\left(1+\sqrt{2}\right)^3}-\frac{1}{\sqrt[3]{\left(1+\sqrt{2}\right)^3}}\)
\(x=1+\sqrt{2}-\frac{1}{1+\sqrt{2}}=\frac{\left(1+\sqrt{2}\right)^2-1}{1+\sqrt{2}}=\frac{2+2\sqrt{2}}{1+\sqrt{2}}\)
x=2
thay vao
x^3+3x-14=0
B=0^2018
B=0
Bài 3: \(3\left(\sqrt{2x^2+1}-1\right)=x\left(1+3x+8\sqrt{2x^2+1}\right)\)
\(\Leftrightarrow\left(3-8x\right)\sqrt{2x^2+1}=3x^2+x+3\)
\(\Rightarrow\left(3-8x\right)^2\left(2x^2+1\right)=\left(3x^2+x+3\right)^2\)
\(\Leftrightarrow119x^4-102x^3+63x^2-54x=0\)
\(\Leftrightarrow x\left(7x-6\right)\left(17x^2+9\right)=0\Rightarrow\orbr{\begin{cases}x=0\\x=\frac{6}{7}\end{cases}}\)
Thử lại, ta nhận được \(x=0\)là nghiệm duy nhất của phương trình
\(\sqrt{a}+\sqrt{b}+\sqrt{c}=3< =>\left(a+b+c+2\sqrt{ab}+2\sqrt{bc}+2\sqrt{ca}\right)=9< =>\sqrt{ab}+\sqrt{bc}+\sqrt{ca}=2\\
\\
\)
Ở đâu có 2 thì thay vào @@
Ta có:
\(\left(\sqrt{a}+\sqrt{b}+\sqrt{c}\right)^2=\left(a+b+c\right)+2\left(\sqrt{ab}+\sqrt{bc}+\sqrt{ca}\right)\)
\(\Rightarrow\sqrt{ab}+\sqrt{bc}+\sqrt{ca}=\frac{\left(\sqrt{a}+\sqrt{b}+\sqrt{c}\right)^2-\left(a+b+c\right)}{2}=\frac{3^2-5}{2}=2\)
Ở đâu có 2 thay bằng \(\sqrt{ab}+\sqrt{bc}+\sqrt{ca}\) là được
Ta có: \(a^2\left(a+1\right)-b^2\left(b-1\right)-11ab+2024\) (1)
Lại có: \(a-b=\sqrt{29+12\sqrt{5}}-2\sqrt{5}\)
\(=\sqrt{\left(2\sqrt{5}\right)^2+2\cdot2\sqrt{5}\cdot3+3^2}-2\sqrt{5}\)
\(=\sqrt{\left(2\sqrt{5}+3\right)^2}-2\sqrt{5}\)
\(=2\sqrt{5}+3-2\sqrt{5}\)
\(=3\)
\(\Rightarrow a=b+3\)
Thay \(a=b+3\) vào (1), ta được:
\(\left(b+3\right)^2\left(b+3+1\right)-b^2\left(b-1\right)-11\left(b+3\right)b+2024\)
\(=\left(b^2+6b+9\right)\left(b+4\right)-b^3+b^2-11b^2-33b+2024\)
\(=b\left(b^2+6b+9\right)+4\left(b^2+6b+9\right)-b^3-10b^2-33b+2024\)
\(=b^3+6b^2+9b+4b^2+24b+36-b^3-10b^2-33b+2024\)
\(=\left(b^3-b^3\right)+\left(6b^2+4b^2-10b^2\right)+\left(9b+24b-33b\right)+\left(2024+36\right)\)
\(=2060\)
$\Rightarrow$ Chọn đáp án $C$.
Ta có : \(a-b=\sqrt{29+12\sqrt{5}}-2\sqrt{5}\)
\(\Rightarrow a-b=\sqrt{20+12\sqrt{5}+9}-2\sqrt{5}\)
\(\Rightarrow a-b=\sqrt{\left(2\sqrt{5}+3\right)^2}-2\sqrt{5}\)
\(\Rightarrow a-b=2\sqrt{5}+3-2\sqrt{5}\)
\(\Rightarrow a-b=3\)
Xét biểu thức : \(a^2\left(a+1\right)-b^2\left(b-1\right)-11ab+2024\)
\(=a^3+a^2-b^3+b^2-11ab+2024\)
\(=a^3-b^3+a^2+b^2-2ab-9ab+2024\)
\(=a^3-b^3-9ab+a^2-2ab+b^2+2024\)
\(=a^3-3ab\left(a-b\right)-b^3+\left(a-b\right)^2+2024\) vì \(a-b=3\)
\(=\left(a-b\right)^3+\left(a-b\right)^2+2024\)
\(=3^3+3^2+2024\)
\(=2060\)
\(\Rightarrow C\)