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)](https://tex.z-dn.net/?f=35x%5E2%2B12x%2B1%3D0%0A%5C%5CD%3D144-140%3D4%0A%5C%5Cx_1%3D%5Cfrac%7B-12%2B2%7D%7B70%7D%3D%5Cfrac%7B-10%7D%7B70%7D%3D%5Cfrac%7B-1%7D%7B7%7D%0A%5C%5Cx_2%3D%5Cfrac%7B-14%7D%7B70%7D%3D%5Cfrac%7B-1%7D%7B5%7D%0A%5C%5C35%28x%2B%5Cfrac%7B1%7D%7B7%7D+%29%28x%2B+%5Cfrac%7B1%7D%7B5%7D+%29%3D%287x%2B1%29%285x%2B1%29%0A%5C%5C%0A%5C%5C35x%5E2%2B48x%2B16%3D0%0A%5C%5CD%3D2304-2240%3D64%0A%5C%5Cx_1%3D%5Cfrac%7B-48%2B8%7D%7B70%7D%3D%5Cfrac%7B-40%7D%7B70%7D%3D-%5Cfrac%7B4%7D%7B7%7D%0A%5C%5Cx_2%3D%5Cfrac%7B-56%7D%7B70%7D%3D-%5Cfrac%7B4%7D%7B5%7D%0A%5C%5C35%28x%2B%5Cfrac%7B4%7D%7B7%7D%29%28x%2B%5Cfrac%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)](https://tex.z-dn.net/?f=%5C%5C35x%5E2%2B27x%2B4%3D0%0A%5C%5CD%3D729-560%3D169%0A%5C%5Cx_1%3D%5Cfrac%7B-27%2B13%7D%7B70%7D%3D%5Cfrac%7B-14%7D%7B70%7D%3D%5Cfrac%7B-1%7D%7B5%7D%0A%5C%5Cx_2%3D%5Cfrac%7B-27-13%7D%7B70%7D%3D%5Cfrac%7B-40%7D%7B70%7D%3D%5Cfrac%7B-4%7D%7B7%7D%0A%5C%5C35%28x%2B%5Cfrac%7B1%7D%7B5%7D%29%28x%2B%5Cfrac%7B4%7D%7B7%7D%29%3D%285x%2B1%29%287x%2B4%29%0A%5C%5C%0A%5C%5C35x%5E2%2B33x%2B4%3D0%0A%5C%5CD%3D1089-560%3D529%0A%5C%5Cx_1%3D%5Cfrac%7B-33%2B23%7D%7B70%7D%3D%5Cfrac%7B-10%7D%7B70%7D%3D%5Cfrac%7B-1%7D%7B7%7D%0A%5C%5Cx_2%3D%5Cfrac%7B-56%7D%7B70%7D%3D-%5Cfrac%7B4%7D%7B5%7D%0A%5C%5C35%28x%2B%5Cfrac%7B1%7D%7B7%7D%29%28x%2B%5Cfrac%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)} ](https://tex.z-dn.net/?f=3%2B%5Csqrt%7B%287x%2B1%29%285x%2B1%29%7D%2B%5Csqrt%7B%285x%2B1%29%287x%2B4%29%7D%3D%5C%5C%0A%3D%5Csqrt%7B%287x%2B1%29%285x%2B4%29%7D%2B%5Csqrt%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}](https://tex.z-dn.net/?f=1%29%5Csqrt%7Bab%7D%3D%5Csqrt%7Ba%7D%2A%5Csqrt%7Bb%7D%0A%5C%5C2%29%5Csqrt%7Bab%7D%3D%5Csqrt%7B-a%7D%2A%5Csqrt%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](https://tex.z-dn.net/?f=3%2B%5Csqrt%7B7x%2B1%7D%2A%5Csqrt%7B5x%2B1%7D%2B%5Csqrt%7B5x%2B1%7D%2A%5Csqrt%7B7x%2B4%7D%5C%5C%3D%5Csqrt%7B7x%2B1%7D%2A%5Csqrt%7B5x%2B4%7D%2B%5Csqrt%7B7x%2B4%7D%2A%5Csqrt%7B5x%2B4%7D%0A%5C%5C%0A%5C%5C3%2B%5Csqrt%7B5x%2B1%7D%2A%28%5Csqrt%7B7x%2B1%7D%2B%5Csqrt%7B7x%2B4%7D%29%3D%5Csqrt%7B5x%2B4%7D%2A%28%5Csqrt%7B7x%2B1%7D%2B%5Csqrt%7B7x%2B4%7D%29%0A%5C%5C3%3D%5Csqrt%7B5x%2B4%7D%2A%28%5Csqrt%7B7x%2B1%7D%2B%5Csqrt%7B7x%2B4%7D%29-%5Csqrt%7B5x%2B1%7D%2A%28%5Csqrt%7B7x%2B1%7D%2B%5Csqrt%7B7x%2B4%7D%29%0A%5C%5C3%3D%28%5Csqrt%7B7x%2B1%7D%2B%5Csqrt%7B7x%2B4%7D%29%2A%28%5Csqrt%7B5x%2B4%7D-%5Csqrt%7B5x%2B1%7D%29%0A%5C%5C%28%5Csqrt%7B7x%2B1%7D%2B%5Csqrt%7B7x%2B4%7D%29%2A%28%5Csqrt%7B5x%2B4%7D-%5Csqrt%7B5x%2B1%7D%29%3D3)
одз:
7x+1>=0
7x+4>=0
5x+4>=0
5x+1>=0
ясно, что
![\sqrt{7x+1}-\sqrt{7x+4} \neq 0](https://tex.z-dn.net/?f=%5Csqrt%7B7x%2B1%7D-%5Csqrt%7B7x%2B4%7D+%5Cneq+0)
поэтому мы можем умножить уравнение на
![(\sqrt{7x+1}-\sqrt{7x+4})](https://tex.z-dn.net/?f=%28%5Csqrt%7B7x%2B1%7D-%5Csqrt%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}](https://tex.z-dn.net/?f=3%2A%28%5Csqrt%7B5x%2B4%7D-%5Csqrt%7B5x%2B1%7D%29%3D3%2A%28%5Csqrt%7B7x%2B4%7D-%5Csqrt%7B7x%2B1%7D%29%0A%5C%5C%5Csqrt%7B5x%2B4%7D-%5Csqrt%7B5x%2B1%7D%3D%5Csqrt%7B7x%2B4%7D-%5Csqrt%7B7x%2B1%7D)
возводим обе части в квадрат:
![5x+4-2\sqrt{(5x+4)(5x+1)}-5x-1= \\=7x+4-2\sqrt{(7x+4)(7x+1)}-7x-1 \\ \\-2\sqrt{(5x+4)(5x+1)}+3=-2\sqrt{(7x+4)(7x+1)}+3 \\\sqrt{(5x+4)(5x+1)}=\sqrt{(7x+4)(7x+1)}](https://tex.z-dn.net/?f=5x%2B4-2%5Csqrt%7B%285x%2B4%29%285x%2B1%29%7D-5x-1%3D%0A%5C%5C%3D7x%2B4-2%5Csqrt%7B%287x%2B4%29%287x%2B1%29%7D-7x-1%0A%5C%5C%0A%5C%5C-2%5Csqrt%7B%285x%2B4%29%285x%2B1%29%7D%2B3%3D-2%5Csqrt%7B%287x%2B4%29%287x%2B1%29%7D%2B3%0A%5C%5C%5Csqrt%7B%285x%2B4%29%285x%2B1%29%7D%3D%5Csqrt%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}](https://tex.z-dn.net/?f=25x%5E2%2B5x%2B20x%2B4%3D49x%5E2%2B7x%2B28x%2B4%0A%5C%5C24x%5E2%2B10x%3D0%0A%5C%5Cx%2824x%2B10%29%3D0%0A%5C%5Cx_1%3D0%0A%5C%5C24x%2B10%3D0%0A%5C%5Cx_2%3D-%5Cfrac%7B10%7D%7B24%7D%3D-%5Cfrac%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](https://tex.z-dn.net/?f=3%2B%5Csqrt%7B-%287x%2B1%29%7D%2A%5Csqrt%7B-%285x%2B1%29%7D%2B%5Csqrt%7B-%285x%2B1%29%7D%2A%5Csqrt%7B-%287x%2B4%29%7D%5C%5C%3D%5Csqrt%7B-%287x%2B1%29%7D%2A%5Csqrt%7B-%285x%2B4%29%7D%2B%5Csqrt%7B-%287x%2B4%29%7D%2A%5Csqrt%7B-%285x%2B4%29%7D%5C%5C%0A%5C%5C3%2B%5Csqrt%7B-%285x%2B1%29%7D%2A%28%5Csqrt%7B-%287x%2B1%29%7D%2B%5Csqrt%7B-%287x%2B4%29%7D%29%3D%0A%5C%5C%3D%5Csqrt%7B-%285x%2B4%29%7D%2A%28%5Csqrt%7B-%287x%2B1%29%7D%2B%5Csqrt%7B-%287x%2B4%29%7D%29%0A%5C%5C%0A%5C%5C%5Csqrt%7B-%285x%2B4%29%7D%2A%28%5Csqrt%7B-%287x%2B1%29%7D%2B%5Csqrt%7B-%287x%2B4%29%7D%29-%0A%5C%5C%5Csqrt%7B-%285x%2B1%29%7D%2A%28%5Csqrt%7B-%287x%2B1%29%7D%2B%5Csqrt%7B-%287x%2B4%29%7D%29%3D3%0A%5C%5C%0A%5C%5C%28%5Csqrt%7B-%285x%2B4%29%7D-%5Csqrt%7B-%285x%2B1%29%7D%29%2A%28%5Csqrt%7B-%287x%2B4%29%7D%2B%5Csqrt%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](https://tex.z-dn.net/?f=%5Csqrt%7B-%287x%2B4%29%7D-%5Csqrt%7B-%287x%2B1%29%7D+%5Cneq+0)
тогда умножаем уравнение на
![(\sqrt{-(7x+4)}-\sqrt{-(7x+1)})](https://tex.z-dn.net/?f=%28%5Csqrt%7B-%287x%2B4%29%7D-%5Csqrt%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)}](https://tex.z-dn.net/?f=%28%5Csqrt%7B-%285x%2B4%29%7D-%5Csqrt%7B-%285x%2B1%29%7D%29%2A%28-7x-4%2B7x%2B1%29%3D+%5C%5C%3D-3%2A%28%5Csqrt%7B-%287x%2B1%29%7D-%5Csqrt%7B-%287x%2B4%29%7D%29+%0A%5C%5C%0A%5C%5C-3%2A%28%5Csqrt%7B-%285x%2B4%29%7D-%5Csqrt%7B-%285x%2B1%29%7D%29%3D-3%2A%28%5Csqrt%7B-%287x%2B4%29%7D-%5Csqrt%7B-%287x%2B1%29%7D%29+%5C%5C%5Csqrt%7B-%285x%2B4%29%7D-%5Csqrt%7B-%285x%2B1%29%7D%3D%5Csqrt%7B-%287x%2B4%29%7D-%5Csqrt%7B-%287x%2B1%29%7D)
выносим
![ \sqrt{-1}](https://tex.z-dn.net/?f=%0A%5Csqrt%7B-1%7D)
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)}](https://tex.z-dn.net/?f=%5Csqrt%7B-1%7D%2A%28%5Csqrt%7B%285x%2B4%29%7D-%5Csqrt%7B%285x%2B1%29%7D%29%3D%5Csqrt%7B-1%7D%2A%28%5Csqrt%7B%287x%2B4%29%7D-%5Csqrt%7B%287x%2B1%29%7D%29%0A%5C%5C%5Csqrt%7B%285x%2B4%29%7D-%5Csqrt%7B%285x%2B1%29%7D%3D%5Csqrt%7B%287x%2B4%29%7D-%5Csqrt%7B%287x%2B1%29%7D)
получили тоже самое уравнение с тем же одз, что и в случае 1
Ответ: x=0