Question

Buba seara va rog frumos, cum as putea dacele exercițiul 4. Va rog frumos.

Answer 1

$\hbox{Avem ecuatia de gr. 2 - g(x)=x^2+ \hbox{bx+c}} \ \ ; \ \hbox{b,c} \in \hbox{R}$

$g^2(x)=f(x) \\\\ (x^2+bx+c)^2=f(x) \\\\\\ f(x)= x^4 +b^2x^2+c^2+2x^3b+2x^2c+2bxc \\\\\\ f(x)=(x+1)(x+2)(x+3)(x+4)+m= \\\\= (x^2+3x+2)(x^2+7x+12)+m= \\\\= x^4+7x^3+12x^2+3x^3+21x^2+36x+2x^2+14x+24+m= \\\\= x^4+10x^3+35x^2+50x+24+m \\\\\\ x^4+10x^3+35x^2+50x+(24+m)=x^4+2bx^3+(b^2+2c)x^2+2bcx+\\ ...+c^2$

[tex]\left\{\begin{matrix} 2b=10\\ b^2+2c=35\\ 2bc=50 \\ 24+m=c^2 \end{matrix}\right. \\\\\\ ===\ \textgreater \ \boxed{b\to 5} \\\\ 2*5*c=50 \ \ \ ===\ \textgreater \ \boxed{c\to 5} \\\\\\ 24+m=5^2 \\\\m=25-24 \\\\ \underline{\underline{m=1}}[/tex]