Question

Rozwiązać przy użyciu wzorów viete'a


[tex]f(x)=3x^2-5x-7\\\\x_{1}^{3} + x_{2}^{3} =\\\\\\|x_{1}-x_{2}|=[/tex]

Answer 1

f(x)=3x^2-5x-7

(X1)^3+(X2)^3=(X1+X2)(X1 ^2-X1X2+X2^2) =

(X1+X2)(X1^2+2X1X2+X2^2-2X1X2-X1X2)=

(X1+X2)((X1+X2)^2-3X1X2)

X1+X2=5/3

X1X2=-7/3

5/3((5/3)^2-3(-7/3)=5/3(25/9+21/3)=5/3(25/9+63/9)=

5/3*88/9=440/27

|X1-X2|=✓(X1-X2)^2=✓x1^2-2x1x2+x2^2=

✓x1^2+2x1x2+x2^2-2x1x2-2x1x2=✓(X1+X2)^2-4x1x2=

✓(5/3)^2-4(-7/3)=✓25/9+28/3=✓25/9+84/9=✓109/9=

✓109/3