Уравнение

[Гост]

[tex]2(a-x)^4-9(a-x)^3(x-b)+14(a-x)^2(x-b)^2-9(a-x)(x-b)^3+2(x-b)^4=0[/tex]
Упътване: Положете [tex]\frac{a-x}{x-b}=y[/tex]

[Xixibg]

1. Проверяваме какво ще се получи за [tex]x=b ; =>(a-x)^4=0 ; =>x=a ; =>x=a=b[/tex] е решение
2. Проверяваме какво ще се получи за [tex]x=a ; =>(x-b)^4=0 ; =>x=b ; =>x=a=b[/tex] е решение
3. Разделяме уравнението на :[tex](a-x)^2(x-b)^2 ; =>\frac{2(a-x)^4}{(a-x)^2(x-b)^2}-\frac{9(a-x)^3(x-b)}{(a-x)^2(x-b)^2}+\frac{14(a-x)^2(x-b)^2}{(a-x)^2(x-b)^2}-\frac{9(a-x)(x-b)^3}{(a-x)^2(x-b)^2}+\frac{2(x-b)^4}{(a-x)^2(x-b)^2}=0[/tex]
[tex]=>\frac{2(a-x)^2}{(x-b)^2}-\frac{9(a-x)}{(x-b)}+14-\frac{9(x-b)}{(a-x)}+\frac{2(x-b)^2}{(a-x)^2}=0[/tex]
Полагаме :[tex]\frac{(a-x)}{(x-b)}+\frac{(x-b)}{(a-x)}=t ; =>\frac{(a-x)^2}{(x-b)^2}+\frac{(x-b)^2}{(a-x)^2}=t^2-2[/tex]
[tex]=>2(t^2-2)-9t+14=0[/tex]
[tex]=>2t^2-9t+10=0[/tex]
[tex]t_1=\frac{5}{2} ; t_2=2[/tex]

I.[tex]\frac{(a-x)}{(x-b)}+\frac{(x-b)}{(a-x)}=\frac{5}{2}[/tex]
[tex]=>(a-x)^2+(x-b)^2=\frac{5}{2}(a-x)(x-b)[/tex]
[tex]x^2-2ax+a^2+x^2-2xb+b^2+\frac{5}{2}x^2-\frac{5}{2}(a+b)x+\frac{5}{2}ab=0[/tex]
[tex]\frac{9}{2}x^2-\frac{9}{2}(a+b)x+a^2+b^2+\frac{5}{2}ab=0[/tex]
[tex]9x^2-9(a+b)x+2a^2+2b^2+5ab=0[/tex]
[tex]D=81a^2+162ab+81b^2-72a^2-72b^2-180ab=9(a-b)^2[/tex]
[tex]x_1=\frac{9(a+b)+3(a-b)}{18}=\frac{2a+b}{3}[/tex]
[tex]x_2=\frac{9(a+b)-3(a-b)}{18}=\frac{a+2b}{3}[/tex]

II.[tex]\frac{(a-x)}{(x-b)}+\frac{(x-b)}{(a-x)}=2[/tex]
[tex]=>(a-x)^2+(x-b)^2=2(a-x)(x-b)[/tex]
[tex]=>(2x-b-a)^2=0[/tex]
[tex]x_3=\frac{a+b}{2}[/tex]