Ответы
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раскладываем трехчлены на множители:
![35x^2+12x+1=0
\D=144-140=4
\x_1=frac{-12+2}{70}=frac{-10}{70}=frac{-1}{7}
\x_2=frac{-14}{70}=frac{-1}{5}
\35(x+frac{1}{7} )(x+ frac{1}{5} )=(7x+1)(5x+1)
\
\35x^2+48x+16=0
\D=2304-2240=64
\x_1=frac{-48+8}{70}=frac{-40}{70}=-frac{4}{7}
\x_2=frac{-56}{70}=-frac{4}{5}
\35(x+frac{4}{7})(x+frac{4}{5})=(7x+4)(5x+4) 35x^2+12x+1=0
\D=144-140=4
\x_1=frac{-12+2}{70}=frac{-10}{70}=frac{-1}{7}
\x_2=frac{-14}{70}=frac{-1}{5}
\35(x+frac{1}{7} )(x+ frac{1}{5} )=(7x+1)(5x+1)
\
\35x^2+48x+16=0
\D=2304-2240=64
\x_1=frac{-48+8}{70}=frac{-40}{70}=-frac{4}{7}
\x_2=frac{-56}{70}=-frac{4}{5}
\35(x+frac{4}{7})(x+frac{4}{5})=(7x+4)(5x+4)](https://tex.z-dn.net/?f=35x%5E2%2B12x%2B1%3D0%0A%5CD%3D144-140%3D4%0A%5Cx_1%3Dfrac%7B-12%2B2%7D%7B70%7D%3Dfrac%7B-10%7D%7B70%7D%3Dfrac%7B-1%7D%7B7%7D%0A%5Cx_2%3Dfrac%7B-14%7D%7B70%7D%3Dfrac%7B-1%7D%7B5%7D%0A%5C35%28x%2Bfrac%7B1%7D%7B7%7D+%29%28x%2B+frac%7B1%7D%7B5%7D+%29%3D%287x%2B1%29%285x%2B1%29%0A%5C%0A%5C35x%5E2%2B48x%2B16%3D0%0A%5CD%3D2304-2240%3D64%0A%5Cx_1%3Dfrac%7B-48%2B8%7D%7B70%7D%3Dfrac%7B-40%7D%7B70%7D%3D-frac%7B4%7D%7B7%7D%0A%5Cx_2%3Dfrac%7B-56%7D%7B70%7D%3D-frac%7B4%7D%7B5%7D%0A%5C35%28x%2Bfrac%7B4%7D%7B7%7D%29%28x%2Bfrac%7B4%7D%7B5%7D%29%3D%287x%2B4%29%285x%2B4%29)
![\35x^2+27x+4=0
\D=729-560=169
\x_1=frac{-27+13}{70}=frac{-14}{70}=frac{-1}{5}
\x_2=frac{-27-13}{70}=frac{-40}{70}=frac{-4}{7}
\35(x+frac{1}{5})(x+frac{4}{7})=(5x+1)(7x+4)
\
\35x^2+33x+4=0
\D=1089-560=529
\x_1=frac{-33+23}{70}=frac{-10}{70}=frac{-1}{7}
\x_2=frac{-56}{70}=-frac{4}{5}
\35(x+frac{1}{7})(x+frac{4}{5})=(7x+1)(5x+4) \35x^2+27x+4=0
\D=729-560=169
\x_1=frac{-27+13}{70}=frac{-14}{70}=frac{-1}{5}
\x_2=frac{-27-13}{70}=frac{-40}{70}=frac{-4}{7}
\35(x+frac{1}{5})(x+frac{4}{7})=(5x+1)(7x+4)
\
\35x^2+33x+4=0
\D=1089-560=529
\x_1=frac{-33+23}{70}=frac{-10}{70}=frac{-1}{7}
\x_2=frac{-56}{70}=-frac{4}{5}
\35(x+frac{1}{7})(x+frac{4}{5})=(7x+1)(5x+4)](https://tex.z-dn.net/?f=%5C35x%5E2%2B27x%2B4%3D0%0A%5CD%3D729-560%3D169%0A%5Cx_1%3Dfrac%7B-27%2B13%7D%7B70%7D%3Dfrac%7B-14%7D%7B70%7D%3Dfrac%7B-1%7D%7B5%7D%0A%5Cx_2%3Dfrac%7B-27-13%7D%7B70%7D%3Dfrac%7B-40%7D%7B70%7D%3Dfrac%7B-4%7D%7B7%7D%0A%5C35%28x%2Bfrac%7B1%7D%7B5%7D%29%28x%2Bfrac%7B4%7D%7B7%7D%29%3D%285x%2B1%29%287x%2B4%29%0A%5C%0A%5C35x%5E2%2B33x%2B4%3D0%0A%5CD%3D1089-560%3D529%0A%5Cx_1%3Dfrac%7B-33%2B23%7D%7B70%7D%3Dfrac%7B-10%7D%7B70%7D%3Dfrac%7B-1%7D%7B7%7D%0A%5Cx_2%3Dfrac%7B-56%7D%7B70%7D%3D-frac%7B4%7D%7B5%7D%0A%5C35%28x%2Bfrac%7B1%7D%7B7%7D%29%28x%2Bfrac%7B4%7D%7B5%7D%29%3D%287x%2B1%29%285x%2B4%29)
получим:
![3+sqrt{(7x+1)(5x+1)}+sqrt{(5x+1)(7x+4)}=\
=sqrt{(7x+1)(5x+4)}+sqrt{(7x+4)(5x+4)}
3+sqrt{(7x+1)(5x+1)}+sqrt{(5x+1)(7x+4)}=\
=sqrt{(7x+1)(5x+4)}+sqrt{(7x+4)(5x+4)}](https://tex.z-dn.net/?f=3%2Bsqrt%7B%287x%2B1%29%285x%2B1%29%7D%2Bsqrt%7B%285x%2B1%29%287x%2B4%29%7D%3D%5C%0A%3Dsqrt%7B%287x%2B1%29%285x%2B4%29%7D%2Bsqrt%7B%287x%2B4%29%285x%2B4%29%7D%0A)
рассматриваем 2 случая:
![1)sqrt{ab}=sqrt{a}*sqrt{b}
\2)sqrt{ab}=sqrt{-a}*sqrt{-b} 1)sqrt{ab}=sqrt{a}*sqrt{b}
\2)sqrt{ab}=sqrt{-a}*sqrt{-b}](https://tex.z-dn.net/?f=1%29sqrt%7Bab%7D%3Dsqrt%7Ba%7D%2Asqrt%7Bb%7D%0A%5C2%29sqrt%7Bab%7D%3Dsqrt%7B-a%7D%2Asqrt%7B-b%7D)
1)
![3+sqrt{7x+1}*sqrt{5x+1}+sqrt{5x+1}*sqrt{7x+4}\=sqrt{7x+1}*sqrt{5x+4}+sqrt{7x+4}*sqrt{5x+4}
\
\3+sqrt{5x+1}*(sqrt{7x+1}+sqrt{7x+4})=sqrt{5x+4}*(sqrt{7x+1}+sqrt{7x+4})
\3=sqrt{5x+4}*(sqrt{7x+1}+sqrt{7x+4})-sqrt{5x+1}*(sqrt{7x+1}+sqrt{7x+4})
\3=(sqrt{7x+1}+sqrt{7x+4})*(sqrt{5x+4}-sqrt{5x+1})
\(sqrt{7x+1}+sqrt{7x+4})*(sqrt{5x+4}-sqrt{5x+1})=3 3+sqrt{7x+1}*sqrt{5x+1}+sqrt{5x+1}*sqrt{7x+4}\=sqrt{7x+1}*sqrt{5x+4}+sqrt{7x+4}*sqrt{5x+4}
\
\3+sqrt{5x+1}*(sqrt{7x+1}+sqrt{7x+4})=sqrt{5x+4}*(sqrt{7x+1}+sqrt{7x+4})
\3=sqrt{5x+4}*(sqrt{7x+1}+sqrt{7x+4})-sqrt{5x+1}*(sqrt{7x+1}+sqrt{7x+4})
\3=(sqrt{7x+1}+sqrt{7x+4})*(sqrt{5x+4}-sqrt{5x+1})
\(sqrt{7x+1}+sqrt{7x+4})*(sqrt{5x+4}-sqrt{5x+1})=3](https://tex.z-dn.net/?f=3%2Bsqrt%7B7x%2B1%7D%2Asqrt%7B5x%2B1%7D%2Bsqrt%7B5x%2B1%7D%2Asqrt%7B7x%2B4%7D%5C%3Dsqrt%7B7x%2B1%7D%2Asqrt%7B5x%2B4%7D%2Bsqrt%7B7x%2B4%7D%2Asqrt%7B5x%2B4%7D%0A%5C%0A%5C3%2Bsqrt%7B5x%2B1%7D%2A%28sqrt%7B7x%2B1%7D%2Bsqrt%7B7x%2B4%7D%29%3Dsqrt%7B5x%2B4%7D%2A%28sqrt%7B7x%2B1%7D%2Bsqrt%7B7x%2B4%7D%29%0A%5C3%3Dsqrt%7B5x%2B4%7D%2A%28sqrt%7B7x%2B1%7D%2Bsqrt%7B7x%2B4%7D%29-sqrt%7B5x%2B1%7D%2A%28sqrt%7B7x%2B1%7D%2Bsqrt%7B7x%2B4%7D%29%0A%5C3%3D%28sqrt%7B7x%2B1%7D%2Bsqrt%7B7x%2B4%7D%29%2A%28sqrt%7B5x%2B4%7D-sqrt%7B5x%2B1%7D%29%0A%5C%28sqrt%7B7x%2B1%7D%2Bsqrt%7B7x%2B4%7D%29%2A%28sqrt%7B5x%2B4%7D-sqrt%7B5x%2B1%7D%29%3D3)
одз:
7x+1>=0
7x+4>=0
5x+4>=0
5x+1>=0
ясно, что
поэтому мы можем умножить уравнение на ![(sqrt{7x+1}-sqrt{7x+4}) (sqrt{7x+1}-sqrt{7x+4})](https://tex.z-dn.net/?f=%28sqrt%7B7x%2B1%7D-sqrt%7B7x%2B4%7D%29)
![3*(sqrt{5x+4}-sqrt{5x+1})=3*(sqrt{7x+4}-sqrt{7x+1})
\sqrt{5x+4}-sqrt{5x+1}=sqrt{7x+4}-sqrt{7x+1} 3*(sqrt{5x+4}-sqrt{5x+1})=3*(sqrt{7x+4}-sqrt{7x+1})
\sqrt{5x+4}-sqrt{5x+1}=sqrt{7x+4}-sqrt{7x+1}](https://tex.z-dn.net/?f=3%2A%28sqrt%7B5x%2B4%7D-sqrt%7B5x%2B1%7D%29%3D3%2A%28sqrt%7B7x%2B4%7D-sqrt%7B7x%2B1%7D%29%0A%5Csqrt%7B5x%2B4%7D-sqrt%7B5x%2B1%7D%3Dsqrt%7B7x%2B4%7D-sqrt%7B7x%2B1%7D)
возводим обе части в квадрат:
![5x+4-2sqrt{(5x+4)(5x+1)}-5x-1=
\=7x+4-2sqrt{(7x+4)(7x+1)}-7x-1
\
\-2sqrt{(5x+4)(5x+1)}+3=-2sqrt{(7x+4)(7x+1)}+3
\sqrt{(5x+4)(5x+1)}=sqrt{(7x+4)(7x+1)} 5x+4-2sqrt{(5x+4)(5x+1)}-5x-1=
\=7x+4-2sqrt{(7x+4)(7x+1)}-7x-1
\
\-2sqrt{(5x+4)(5x+1)}+3=-2sqrt{(7x+4)(7x+1)}+3
\sqrt{(5x+4)(5x+1)}=sqrt{(7x+4)(7x+1)}](https://tex.z-dn.net/?f=5x%2B4-2sqrt%7B%285x%2B4%29%285x%2B1%29%7D-5x-1%3D%0A%5C%3D7x%2B4-2sqrt%7B%287x%2B4%29%287x%2B1%29%7D-7x-1%0A%5C%0A%5C-2sqrt%7B%285x%2B4%29%285x%2B1%29%7D%2B3%3D-2sqrt%7B%287x%2B4%29%287x%2B1%29%7D%2B3%0A%5Csqrt%7B%285x%2B4%29%285x%2B1%29%7D%3Dsqrt%7B%287x%2B4%29%287x%2B1%29%7D)
еще раз возводим в квадрат:
(5x+4)(5x+1)=(7x+4)(7x+1)
но:
(5x+4)(5x+1)>=0 и (7x+4)(7x+1)>=0
![25x^2+5x+20x+4=49x^2+7x+28x+4
\24x^2+10x=0
\x(24x+10)=0
\x_1=0
\24x+10=0
\x_2=-frac{10}{24}=-frac{5}{12} 25x^2+5x+20x+4=49x^2+7x+28x+4
\24x^2+10x=0
\x(24x+10)=0
\x_1=0
\24x+10=0
\x_2=-frac{10}{24}=-frac{5}{12}](https://tex.z-dn.net/?f=25x%5E2%2B5x%2B20x%2B4%3D49x%5E2%2B7x%2B28x%2B4%0A%5C24x%5E2%2B10x%3D0%0A%5Cx%2824x%2B10%29%3D0%0A%5Cx_1%3D0%0A%5C24x%2B10%3D0%0A%5Cx_2%3D-frac%7B10%7D%7B24%7D%3D-frac%7B5%7D%7B12%7D)
проверяем:
(5*(-5)/12+4)(5*(-5/12)+1)>=0
-25/12<1, значит 2 скобка меньше 0.
корень x=-5/12 не подходит.
проверяем 0:
4*1>=0
4*1>=0
1>=0
4>=0
4>=0
1>=0 - верно, значит x=0 - корень уравнения
теперь рассмотрим 2 случай:
2)
![3+sqrt{-(7x+1)}*sqrt{-(5x+1)}+sqrt{-(5x+1)}*sqrt{-(7x+4)}\=sqrt{-(7x+1)}*sqrt{-(5x+4)}+sqrt{-(7x+4)}*sqrt{-(5x+4)}\
\3+sqrt{-(5x+1)}*(sqrt{-(7x+1)}+sqrt{-(7x+4)})=
\=sqrt{-(5x+4)}*(sqrt{-(7x+1)}+sqrt{-(7x+4)})
\
\sqrt{-(5x+4)}*(sqrt{-(7x+1)}+sqrt{-(7x+4)})-
\sqrt{-(5x+1)}*(sqrt{-(7x+1)}+sqrt{-(7x+4)})=3
\
\(sqrt{-(5x+4)}-sqrt{-(5x+1)})*(sqrt{-(7x+4)}+sqrt{-(7x+1)})=3 3+sqrt{-(7x+1)}*sqrt{-(5x+1)}+sqrt{-(5x+1)}*sqrt{-(7x+4)}\=sqrt{-(7x+1)}*sqrt{-(5x+4)}+sqrt{-(7x+4)}*sqrt{-(5x+4)}\
\3+sqrt{-(5x+1)}*(sqrt{-(7x+1)}+sqrt{-(7x+4)})=
\=sqrt{-(5x+4)}*(sqrt{-(7x+1)}+sqrt{-(7x+4)})
\
\sqrt{-(5x+4)}*(sqrt{-(7x+1)}+sqrt{-(7x+4)})-
\sqrt{-(5x+1)}*(sqrt{-(7x+1)}+sqrt{-(7x+4)})=3
\
\(sqrt{-(5x+4)}-sqrt{-(5x+1)})*(sqrt{-(7x+4)}+sqrt{-(7x+1)})=3](https://tex.z-dn.net/?f=3%2Bsqrt%7B-%287x%2B1%29%7D%2Asqrt%7B-%285x%2B1%29%7D%2Bsqrt%7B-%285x%2B1%29%7D%2Asqrt%7B-%287x%2B4%29%7D%5C%3Dsqrt%7B-%287x%2B1%29%7D%2Asqrt%7B-%285x%2B4%29%7D%2Bsqrt%7B-%287x%2B4%29%7D%2Asqrt%7B-%285x%2B4%29%7D%5C%0A%5C3%2Bsqrt%7B-%285x%2B1%29%7D%2A%28sqrt%7B-%287x%2B1%29%7D%2Bsqrt%7B-%287x%2B4%29%7D%29%3D%0A%5C%3Dsqrt%7B-%285x%2B4%29%7D%2A%28sqrt%7B-%287x%2B1%29%7D%2Bsqrt%7B-%287x%2B4%29%7D%29%0A%5C%0A%5Csqrt%7B-%285x%2B4%29%7D%2A%28sqrt%7B-%287x%2B1%29%7D%2Bsqrt%7B-%287x%2B4%29%7D%29-%0A%5Csqrt%7B-%285x%2B1%29%7D%2A%28sqrt%7B-%287x%2B1%29%7D%2Bsqrt%7B-%287x%2B4%29%7D%29%3D3%0A%5C%0A%5C%28sqrt%7B-%285x%2B4%29%7D-sqrt%7B-%285x%2B1%29%7D%29%2A%28sqrt%7B-%287x%2B4%29%7D%2Bsqrt%7B-%287x%2B1%29%7D%29%3D3)
одз:
-(7x+1)<=0
-(7x+4)<=0
-(5x+4)<=0
-(5x+1)<=0
или
7x+1>=0
7x+4>=0
5x+4>=0
5x+1>=0
одз 1 и 2 случая совпали
ясно, что![sqrt{-(7x+4)}-sqrt{-(7x+1)} neq 0 sqrt{-(7x+4)}-sqrt{-(7x+1)} neq 0](https://tex.z-dn.net/?f=sqrt%7B-%287x%2B4%29%7D-sqrt%7B-%287x%2B1%29%7D+neq+0)
тогда умножаем уравнение на![(sqrt{-(7x+4)}-sqrt{-(7x+1)}) (sqrt{-(7x+4)}-sqrt{-(7x+1)})](https://tex.z-dn.net/?f=%28sqrt%7B-%287x%2B4%29%7D-sqrt%7B-%287x%2B1%29%7D%29)
![(sqrt{-(5x+4)}-sqrt{-(5x+1)})*(-7x-4+7x+1)= \=-3*(sqrt{-(7x+1)}-sqrt{-(7x+4)})
\
\-3*(sqrt{-(5x+4)}-sqrt{-(5x+1)})=-3*(sqrt{-(7x+4)}-sqrt{-(7x+1)}) \sqrt{-(5x+4)}-sqrt{-(5x+1)}=sqrt{-(7x+4)}-sqrt{-(7x+1)} (sqrt{-(5x+4)}-sqrt{-(5x+1)})*(-7x-4+7x+1)= \=-3*(sqrt{-(7x+1)}-sqrt{-(7x+4)})
\
\-3*(sqrt{-(5x+4)}-sqrt{-(5x+1)})=-3*(sqrt{-(7x+4)}-sqrt{-(7x+1)}) \sqrt{-(5x+4)}-sqrt{-(5x+1)}=sqrt{-(7x+4)}-sqrt{-(7x+1)}](https://tex.z-dn.net/?f=%28sqrt%7B-%285x%2B4%29%7D-sqrt%7B-%285x%2B1%29%7D%29%2A%28-7x-4%2B7x%2B1%29%3D+%5C%3D-3%2A%28sqrt%7B-%287x%2B1%29%7D-sqrt%7B-%287x%2B4%29%7D%29+%0A%5C%0A%5C-3%2A%28sqrt%7B-%285x%2B4%29%7D-sqrt%7B-%285x%2B1%29%7D%29%3D-3%2A%28sqrt%7B-%287x%2B4%29%7D-sqrt%7B-%287x%2B1%29%7D%29+%5Csqrt%7B-%285x%2B4%29%7D-sqrt%7B-%285x%2B1%29%7D%3Dsqrt%7B-%287x%2B4%29%7D-sqrt%7B-%287x%2B1%29%7D)
выносим
за скобку и сокращаем:
![sqrt{-1}*(sqrt{(5x+4)}-sqrt{(5x+1)})=sqrt{-1}*(sqrt{(7x+4)}-sqrt{(7x+1)})
\sqrt{(5x+4)}-sqrt{(5x+1)}=sqrt{(7x+4)}-sqrt{(7x+1)} sqrt{-1}*(sqrt{(5x+4)}-sqrt{(5x+1)})=sqrt{-1}*(sqrt{(7x+4)}-sqrt{(7x+1)})
\sqrt{(5x+4)}-sqrt{(5x+1)}=sqrt{(7x+4)}-sqrt{(7x+1)}](https://tex.z-dn.net/?f=sqrt%7B-1%7D%2A%28sqrt%7B%285x%2B4%29%7D-sqrt%7B%285x%2B1%29%7D%29%3Dsqrt%7B-1%7D%2A%28sqrt%7B%287x%2B4%29%7D-sqrt%7B%287x%2B1%29%7D%29%0A%5Csqrt%7B%285x%2B4%29%7D-sqrt%7B%285x%2B1%29%7D%3Dsqrt%7B%287x%2B4%29%7D-sqrt%7B%287x%2B1%29%7D)
получили тоже самое уравнение с тем же одз, что и в случае 1
Ответ: x=0
получим:
рассматриваем 2 случая:
1)
одз:
7x+1>=0
7x+4>=0
5x+4>=0
5x+1>=0
ясно, что
возводим обе части в квадрат:
еще раз возводим в квадрат:
(5x+4)(5x+1)=(7x+4)(7x+1)
но:
(5x+4)(5x+1)>=0 и (7x+4)(7x+1)>=0
проверяем:
(5*(-5)/12+4)(5*(-5/12)+1)>=0
-25/12<1, значит 2 скобка меньше 0.
корень x=-5/12 не подходит.
проверяем 0:
4*1>=0
4*1>=0
1>=0
4>=0
4>=0
1>=0 - верно, значит x=0 - корень уравнения
теперь рассмотрим 2 случай:
2)
одз:
-(7x+1)<=0
-(7x+4)<=0
-(5x+4)<=0
-(5x+1)<=0
или
7x+1>=0
7x+4>=0
5x+4>=0
5x+1>=0
одз 1 и 2 случая совпали
ясно, что
тогда умножаем уравнение на
выносим
получили тоже самое уравнение с тем же одз, что и в случае 1
Ответ: x=0
Ответ дал:
0
Не очень читаемое (((
Ответ дал:
0
++
Ответ дал:
0
зато правильно
Ответ дал:
0
Полагаю есть проще решение :)
Ответ дал:
0
Корень из произведения не всегда равен произведению корней
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