Nung nóng topic nào:
Bài 81 Giải phương trình: $x^2+\sqrt{x}=5$
Bài này ra nghiệm lẻ. Nhưng vẫn mong "mậy người" tìm ra cách giải
Ồ, bài này chưa ai giải !!!
ĐKXĐ: $x \geq 0$
Đặt $\sqrt{x}=y \geq 0$
Suy ra $y^4+y-5=0$
Hay $y^4+2my^2+m^2=2my^2-y+m^2+5$
Ta sẽ tìm $m > 0$ để $2my^2-y+m^2+5$ là bình phương một số
Hay $\Delta =0$
Hay $1-8m(m^2+5)=0$
Hay $8m^3+40m-1=0$
Đặt $m=\frac{t}{12}-\frac{20}{t}$
Suy ra $t^6-216t^3-13824000=0$
Hay $t=\pm \sqrt[3]{108+12\sqrt{96081}}$
Mà để $m>0$ suy ra $t= \sqrt[3]{108+12\sqrt{96081}}$
Vậy $m=\frac{\sqrt[3]{108+12\sqrt{96081}}}{12}-\frac{20}{\sqrt[3]{108+12\sqrt{96081}}}$
Suy ra $(y^2+\frac{\sqrt[3]{108+12\sqrt{96081}}}{12}-\frac{20}{\sqrt[3]{108+12\sqrt{96081}}})^2=-{\frac {1}{59719680000}}\,\sqrt [3]{108+12\,\sqrt {96081}} \left( -
4800-9\,\sqrt [3]{108+12\,\sqrt {96081}}+\sqrt [3]{108+12\,\sqrt {
96081}}\sqrt {96081} \right) \left( -9\,\sqrt [3]{108+12\,\sqrt {
96081}}+20\, \left( 108+12\,\sqrt {96081} \right) ^{2/3}+4800+\sqrt [3
]{108+12\,\sqrt {96081}}\sqrt {96081}-1440\,x \right) ^{2}$
Hay $((y^2+\frac{\sqrt[3]{108+12\sqrt{96081}}}{12}-\frac{20}{\sqrt[3]{108+12\sqrt{96081}}})-({\frac {1}{345600}}\,\sqrt {2}\sqrt {\sqrt [3]{108+12\,\sqrt {96081}}
\left( 4800+9\,\sqrt [3]{108+12\,\sqrt {96081}}-\sqrt [3]{108+12\,
\sqrt {96081}}\sqrt {96081} \right) }
(-9\,\sqrt [3]{108+12\,\sqrt {96081}}+20\, \left( 108+12\,\sqrt {96081}
\right) ^{2/3}+4800+\sqrt [3]{108+12\,\sqrt {96081}}\sqrt {96081}-
1440\,x)))
((y^2+\frac{\sqrt[3]{108+12\sqrt{96081}}}{12}-\frac{20}{\sqrt[3]{108+12\sqrt{96081}}})+({\frac {1}{345600}}\,\sqrt {2}\sqrt {\sqrt [3]{108+12\,\sqrt {96081}}
\left( 4800+9\,\sqrt [3]{108+12\,\sqrt {96081}}-\sqrt [3]{108+12\,
\sqrt {96081}}\sqrt {96081} \right) }
(-9\,\sqrt [3]{108+12\,\sqrt {96081}}+20\, \left( 108+12\,\sqrt {96081}
\right) ^{2/3}+4800+\sqrt [3]{108+12\,\sqrt {96081}}\sqrt {96081}-
1440\,x)))$
Suy ra
${y}^{2}+{\frac {1}{240}}\,\sqrt {2}\sqrt {\sqrt [3]{108+12\,\sqrt {
96081}} \left( 4800+9\,\sqrt [3]{108+12\,\sqrt {96081}}-\sqrt [3]{108+
12\,\sqrt {96081}}\sqrt {96081} \right) }y+1/12\,\sqrt [3]{108+12\,
\sqrt {96081}}-20\,{\frac {1}{\sqrt [3]{108+12\,\sqrt {96081}}}}-{
\frac {1}{345600}}\,\sqrt {2}\sqrt {\sqrt [3]{108+12\,\sqrt {96081}}
\left( 4800+9\,\sqrt [3]{108+12\,\sqrt {96081}}-\sqrt [3]{108+12\,
\sqrt {96081}}\sqrt {96081} \right) } \left( -9\,\sqrt [3]{108+12\,
\sqrt {96081}}+20\, \left( 108+12\,\sqrt {96081} \right) ^{2/3}+4800+
\sqrt [3]{108+12\,\sqrt {96081}}\sqrt {96081} \right) =0$
Hoặc:
${y}^{2}-{\frac {1}{240}}\,\sqrt {2}\sqrt {\sqrt [3]{108+12\,\sqrt {
96081}} \left( 4800+9\,\sqrt [3]{108+12\,\sqrt {96081}}-\sqrt [3]{108+
12\,\sqrt {96081}}\sqrt {96081} \right) }y+1/12\,\sqrt [3]{108+12\,
\sqrt {96081}}-20\,{\frac {1}{\sqrt [3]{108+12\,\sqrt {96081}}}}+{
\frac {1}{345600}}\,\sqrt {2}\sqrt {\sqrt [3]{108+12\,\sqrt {96081}}
\left( 4800+9\,\sqrt [3]{108+12\,\sqrt {96081}}-\sqrt [3]{108+12\,
\sqrt {96081}}\sqrt {96081} \right) } \left( -9\,\sqrt [3]{108+12\,
\sqrt {96081}}+20\, \left( 108+12\,\sqrt {96081} \right) ^{2/3}+4800+
\sqrt [3]{108+12\,\sqrt {96081}}\sqrt {96081} \right) =0$
Xét PT đầu là phương trình bậc 2 ẩn $y$ mà $y \geq 0$ nên ta tìm được:
$y=-{\frac {1}{480}}\,{\frac {\sqrt {2}\sqrt {\sqrt [3]{108+12\,\sqrt {
96081}} \left( 4800+9\,\sqrt [3]{108+12\,\sqrt {96081}}-\sqrt [3]{108+
12\,\sqrt {96081}}\sqrt {96081} \right) }\sqrt [3]{108+12\,\sqrt {
96081}}-4\,\sqrt {-64800+144000\,\sqrt [3]{108+12\,\sqrt {96081}}-7200
\,\sqrt {96081}+48000\,\sqrt {2}\sqrt {\sqrt [3]{108+12\,\sqrt {96081}
} \left( 4800+9\,\sqrt [3]{108+12\,\sqrt {96081}}-\sqrt [3]{108+12\,
\sqrt {96081}}\sqrt {96081} \right) }+90\,\sqrt {2}\sqrt {\sqrt [3]{
108+12\,\sqrt {96081}} \left( 4800+9\,\sqrt [3]{108+12\,\sqrt {96081}}
-\sqrt [3]{108+12\,\sqrt {96081}}\sqrt {96081} \right) }\sqrt [3]{108+
12\,\sqrt {96081}}+10\,\sqrt {2}\sqrt {\sqrt [3]{108+12\,\sqrt {96081}
} \left( 4800+9\,\sqrt [3]{108+12\,\sqrt {96081}}-\sqrt [3]{108+12\,
\sqrt {96081}}\sqrt {96081} \right) }\sqrt [3]{108+12\,\sqrt {96081}}
\sqrt {96081}+200\,\sqrt {2}\sqrt {\sqrt [3]{108+12\,\sqrt {96081}}
\left( 4800+9\,\sqrt [3]{108+12\,\sqrt {96081}}-\sqrt [3]{108+12\,
\sqrt {96081}}\sqrt {96081} \right) } \left( 108+12\,\sqrt {96081}
\right) ^{2/3}}}{\sqrt [3]{108+12\,\sqrt {96081}}}}$
Suy ra $x=y^2={\frac {1}{230400}}\,{\frac { \left( \sqrt {2}\sqrt {\sqrt [3]{108+12
\,\sqrt {96081}} \left( 4800+9\,\sqrt [3]{108+12\,\sqrt {96081}}-
\sqrt [3]{108+12\,\sqrt {96081}}\sqrt {96081} \right) }\sqrt [3]{108+
12\,\sqrt {96081}}-4\,\sqrt {-64800+144000\,\sqrt [3]{108+12\,\sqrt {
96081}}-7200\,\sqrt {96081}+48000\,\sqrt {2}\sqrt {\sqrt [3]{108+12\,
\sqrt {96081}} \left( 4800+9\,\sqrt [3]{108+12\,\sqrt {96081}}-\sqrt [
3]{108+12\,\sqrt {96081}}\sqrt {96081} \right) }+90\,\sqrt {2}\sqrt {
\sqrt [3]{108+12\,\sqrt {96081}} \left( 4800+9\,\sqrt [3]{108+12\,
\sqrt {96081}}-\sqrt [3]{108+12\,\sqrt {96081}}\sqrt {96081} \right) }
\sqrt [3]{108+12\,\sqrt {96081}}+10\,\sqrt {2}\sqrt {\sqrt [3]{108+12
\,\sqrt {96081}} \left( 4800+9\,\sqrt [3]{108+12\,\sqrt {96081}}-
\sqrt [3]{108+12\,\sqrt {96081}}\sqrt {96081} \right) }\sqrt [3]{108+
12\,\sqrt {96081}}\sqrt {96081}+200\,\sqrt {2}\sqrt {\sqrt [3]{108+12
\,\sqrt {96081}} \left( 4800+9\,\sqrt [3]{108+12\,\sqrt {96081}}-
\sqrt [3]{108+12\,\sqrt {96081}}\sqrt {96081} \right) } \left( 108+12
\,\sqrt {96081} \right) ^{2/3}} \right) ^{2}}{ \left( 108+12\,\sqrt {
96081} \right) ^{2/3}}}$
Xét phương trình bậc 2 ở dưới ta thấy: phương trình không có nghiệm dương nên vô lý
Vậy phương trình đã cho có một nghiệm
$$x={\frac {1}{230400}}\,{\frac { \left( \sqrt {2}\sqrt {\sqrt [3]{108+12
\,\sqrt {96081}} \left( 4800+9\,\sqrt [3]{108+12\,\sqrt {96081}}-
\sqrt [3]{108+12\,\sqrt {96081}}\sqrt {96081} \right) }\sqrt [3]{108+
12\,\sqrt {96081}}-4\,\sqrt {-64800+144000\,\sqrt [3]{108+12\,\sqrt {
96081}}-7200\,\sqrt {96081}+48000\,\sqrt {2}\sqrt {\sqrt [3]{108+12\,
\sqrt {96081}} \left( 4800+9\,\sqrt [3]{108+12\,\sqrt {96081}}-\sqrt [
3]{108+12\,\sqrt {96081}}\sqrt {96081} \right) }+90\,\sqrt {2}\sqrt {
\sqrt [3]{108+12\,\sqrt {96081}} \left( 4800+9\,\sqrt [3]{108+12\,
\sqrt {96081}}-\sqrt [3]{108+12\,\sqrt {96081}}\sqrt {96081} \right) }
\sqrt [3]{108+12\,\sqrt {96081}}+10\,\sqrt {2}\sqrt {\sqrt [3]{108+12
\,\sqrt {96081}} \left( 4800+9\,\sqrt [3]{108+12\,\sqrt {96081}}-
\sqrt [3]{108+12\,\sqrt {96081}}\sqrt {96081} \right) }\sqrt [3]{108+
12\,\sqrt {96081}}\sqrt {96081}+200\,\sqrt {2}\sqrt {\sqrt [3]{108+12
\,\sqrt {96081}} \left( 4800+9\,\sqrt [3]{108+12\,\sqrt {96081}}-
\sqrt [3]{108+12\,\sqrt {96081}}\sqrt {96081} \right) } \left( 108+12
\,\sqrt {96081} \right) ^{2/3}} \right) ^{2}}{ \left( 108+12\,\sqrt {
96081} \right) ^{2/3}}}$$
__________________________
P/s: Nếu muốn xem gọn hơn thì như sau:
$$x=\left( -1/12\,\sqrt {6}\sqrt {{\frac { \left( 108+12\,\sqrt {9681}
\right) ^{2/3}-240}{\sqrt [3]{108+12\,\sqrt {96081}}}}}+1/12\,\sqrt {
6}\sqrt {{\frac {-\sqrt {{\frac { \left( 108+12\,\sqrt {96081}
\right) ^{2/3}-240}{\sqrt [3]{108+12\,\sqrt {96081}}}}} \left( 108+12
\,\sqrt {96081} \right) ^{2/3}+240\,\sqrt {{\frac { \left( 108+12\,
\sqrt {96081} \right) ^{2/3}-240}{\sqrt [3]{108+12\,\sqrt {96081}}}}}+
12\,\sqrt {6}\sqrt [3]{108+12\,\sqrt {96081}}}{\sqrt [3]{108+12\,
\sqrt {96081}}\sqrt {{\frac { \left( 108+12\,\sqrt {96081} \right) ^{2
/3}-240}{\sqrt [3]{108+12\,\sqrt {96081}}}}}}}} \right) ^{2}$$