tìm \(n\in N,y\in Z\)sao cho \(2^n+3=y^2\)
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Dat \(x+y=t;xy=v\left(t,v\ne0\right)\)
HPT tro thanh
\(\hept{\begin{cases}t+\frac{t}{v}=\frac{9}{2}\\v+\frac{1}{v}=\frac{5}{2}\end{cases}}\)
\(\Leftrightarrow\hept{\begin{cases}t+\frac{t}{v}=\frac{9}{2}\left(1\right)\\v^2-\frac{5}{2}v+1=0\left(2\right)\end{cases}}\)
Xet (2):
\(\Delta=\frac{25}{4}-4=\frac{9}{4}\)
Suy ra:
\(v_1=4;v_2=1\)
Voi \(v=4\)thi thay vao HPT thay khong thoa man nen loai
Voi \(v=1\)thay vao HPT thay khong thoa man nen loai
Vay HPT vo nghiem
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a) \(\sqrt{x-1}+\sqrt{2x-1}=5\)
\(\Leftrightarrow3x-2+2\sqrt{\left(x-1\right)\left(2x-1\right)}=25\)
\(\Leftrightarrow2\sqrt{\left(x-1\right)\left(2x-1\right)}=25-3x+2\)
\(\Leftrightarrow2\sqrt{\left(x-1\right)\left(2x-1\right)}=-3x+27\)
Bình phương 2 vế, ta được:
\(\Leftrightarrow4\left(x-1\right)\left(2x-1\right)=9\left(x-9\right)^2\)
\(\Leftrightarrow8x^2-4x-8x+4=9x^2-162x+729\)
\(\Leftrightarrow8x^2-12x+4-9x^2+162x-729=0\)
\(\Leftrightarrow-x^2+150x-725=0\)
\(\Leftrightarrow x^2-150x+725=0\)
\(\Leftrightarrow\left(x-145\right)\left(x-5\right)=0\)
\(\Leftrightarrow\orbr{\begin{cases}x-145=0\\x-5=0\end{cases}}\Leftrightarrow\orbr{\begin{cases}x=145\left(ktm\right)\\x=5\left(tm\right)\end{cases}}\)
\(\Rightarrow x=5\)
b) \(x+\sqrt{2x-1}-2=0\)
\(\Leftrightarrow\sqrt{2x-1}=2-x\)
Bình phương 2 vế, ta được:
\(\Leftrightarrow2x-1=4-4x^2+x^2=0\)
\(\Leftrightarrow2x-1-4+4x-x^2=0\)
\(\Leftrightarrow6x-5-x^2=0\)
\(\Leftrightarrow\left(x-5\right)\left(x-1\right)=0\)
\(\Leftrightarrow\orbr{\begin{cases}x-5=0\\x-1=0\end{cases}}\Leftrightarrow\orbr{\begin{cases}x=5\left(ktm\right)\\x=1\left(tm\right)\end{cases}}\)