Đặt :

\(x^2=t\) => t >0

Phương trình tương đương :

\(t^2+2mt+4=0\) (*)

Để phương trình trên co 4 nghiệm phân biệt thì (*) phải có 2 nghiệm dương

=>| Điều kiện :

\(\Delta'=m-4>0\)

\(\Rightarrow m>4\)

Theo hệ thức Vi-ét :

\(\left\{{}\begin{matrix}x_1+x_2+x_3+x_4=-\dfrac{b}{a}=0\\x_1x_2+x_1x_3+x_1x_4+x_2x_3+x_2x_4=\dfrac{c}{a}=m\\x_1x_2x_3+x_1x_2x_4+x_1x_3x_4+x_2x_3x_4=-\dfrac{d}{a}=0\\x_1x_2x_3x_4=\dfrac{e}{a}=4\end{matrix}\right.\)

Mũ 4 phương trình đầu tiên lên rồi áp vào

\(x_1^4+x_2^4+x_3^4+x^4_4=32\) , sử dụng các phương trình bên dưới nữa để giải ra m là được

Đặt \(x^2=t\ge0\) pt trở thành: \(t^2-2\left(m-3\right)t-2m-24=0\) (1)

Để pt đã cho có 4 nghiệm pb thì (1) có 2 nghiệm dương pb

\(\Leftrightarrow\left\{{}\begin{matrix}\Delta'=m^2-4m+33>0\\t_1+t_2=2\left(m-3\right)>0\\t_1t_2=-2m-24>0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}m>3\\m< -12\end{matrix}\right.\)

Không tồn tại m thỏa mãn yêu cầu đề bài

1.You will be healthier if you consume __________ fruits and vegetables.A. moreB. lessC. fewerD. little2.I have a temperature, _________.A. but I go to bed earlyB. so I feel tiredC. or I am putting on weightD. and I eat more vegetables3.My dad enjoys ________ his bike to work.A. to rideB. rideC. ridesD. riding4.I ________ making models interesting, but my sister doesn't.A. doesB. thinkC. findD. say5.My father loved _________ horse-riding when he lived in France.A. doingB. goingC. playingD. taking6.Minh_________100 postcards so far. He intends to sell them to raise funds.A. makesB. madeC. has madeD. make7.Ben __________ three eggshells and he will continue the fourth one.A. carvesB. has carvedC. carvedD. will carved8.David is short-sighted. He should spend_________time staring at his smart phone.A. moreB. lessC. fewerD. few9.I think, in the future robots__________able to feel emotions like humans.A. areB. have beenC. wereD. will be10.You should eat less chocolate, _________ you can put on weight.A. andB. orC. butD. so

1.You will be healthier if you consume __________ fruits and vegetables.A. moreB. lessC. fewerD. little2.I have a temperature, _________.A. but I go to bed earlyB. so I feel tiredC. or I am putting on weightD. and I eat more vegetables3.My dad enjoys ________ his bike to work.A. to rideB. rideC. ridesD. riding4.I ________ making models interesting, but my sister doesn't.A. doesB. thinkC. findD. say5.My father loved _________ horse-riding when he lived in France.A. doingB. goingC. playingD. taking6.Minh_________100 postcards so far. He intends to sell them to raise funds.A. makesB. madeC. has madeD. make7.Ben __________ three eggshells and he will continue the fourth one.A. carvesB. has carvedC. carvedD. will carved8.David is short-sighted. He should spend_________time staring at his smart phone.A. moreB. lessC. fewerD. few9.I think, in the future robots__________able to feel emotions like humans.A. areB. have beenC. wereD. will be10.You should eat less chocolate, _________ you can put on weight.A. andB. orC. butD. so

1.You will be healthier if you consume __________ fruits and vegetables.A. moreB. lessC. fewerD. little2.I have a temperature, _________.A. but I go to bed earlyB. so I feel tiredC. or I am putting on weightD. and I eat more vegetables3.My dad enjoys ________ his bike to work.A. to rideB. rideC. ridesD. riding4.I ________ making models interesting, but my sister doesn't.A. doesB. thinkC. findD. say5.My father loved _________ horse-riding when he lived in France.A. doingB. goingC. playingD. taking6.Minh_________100 postcards so far. He intends to sell them to raise funds.A. makesB. madeC. has madeD. make7.Ben __________ three eggshells and he will continue the fourth one.A. carvesB. has carvedC. carvedD. will carved8.David is short-sighted. He should spend_________time staring at his smart phone.A. moreB. lessC. fewerD. few9.I think, in the future robots__________able to feel emotions like humans.A. areB. have beenC. wereD. will be10.You should eat less chocolate, _________ you can put on weight.A. andB. orC. butD. so

Theo định lí Viet \(\left\{{}\begin{matrix}x_1+x_2=5\\x_1x_2=-1\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}y_1+y_2=x_1^4+x_2^4=\left[\left(x_1+x_2\right)^2-2x_1x_2\right]^2-2x_1^2x_2^2=727\\y_1y_2=x_1^4x_2^4=1\end{matrix}\right.\)

Phương trình cần tìm có dạng \(ax^2+bx+c=0\left(1\right)\)

\(\Rightarrow\left\{{}\begin{matrix}-\dfrac{b}{a}=727\\\dfrac{c}{a}=1\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}b=-727a\\c=a\end{matrix}\right.\)

\(\left(1\right)\Leftrightarrow ax^2-727ax+a=0\)

\(\Leftrightarrow x^2-727x+1=0\)

Ta có

\(x^4+y^4=7z^4+5\Leftrightarrow x^4+y^4+z^4=8z^4+5\)

Áp dụng tính chất lũy thừa bậc 4 của số nguyên a khi chia cho 8 dư 0 hoặc 1

tức là \(a^4\equiv0,1\left(mod8\right)\)

\(\Rightarrow a^4+b^4+c^4\equiv0,1,2,3\left(mod8\right)\)

Mà \(8z^4+5\equiv5\left(mod8\right)\)

vậy pt k có nghiệm nguyên