Question

设m n是一元二次方程ax平方+bx+c=0的两个根,求代数式a(m立方+n立方）+b（m平方+n平方）+c（m+n）的

Answer 1

m + n = -b/a

m X n = c/a

=a(m^3 + n^3) + b(m^2 + n^2) + c(m + n)

=a[(m+n)(m^2-mn+n^2)]+(m+n)^2-2mn+c(m+n)

=a{(m+n)[(m+n)^2-3mn]}+(m+n)^2-2mn+c(m+n)

=a{(-b/a)[(-b/a)^2-3c/a]}+(-b/a)^2

-2c/a+c(-b/a)

=(3abc-b^3)/(a^2) + (b^2)/(a^2) -

(2ac)/(a^2)-abc/(a^2)

=(3abc-b^3+b^2-2ac-abc)/(a^2)

=(2abc-b^3+b^2-2ac)/(a^2)