Given three quadratic equations in terms of x,

ax^2+bx+c = 0

bx^2+cx+a = 0


share a same real root. Find the value of \dfrac{a^2}{bc}+\dfrac{b^2}{ac}+\dfrac{c^2}{ab}

Collected in the board: Quadratic function

Steven Zheng posted 5 months ago


Let x be the shared root of the three quadratic equations.

Addition of the three equations gives


Evaluate the discriminant of the quadratic function y=x^2+x+1

\Delta = 1^2-4\cdot 1\cdot 1< 0

Therefore, x^2+x+1>0

Then, another factor


c = -(a+b)

Using the equation (2), now we can determine the expression




Apply the cube of binomila identity

(a+b)^3 = a^3+b^3+3ab(a+b)

a^3+b^3-(a+b)^3 =-3ab(a+b)



=\dfrac{-3ab(a+b)}{abc} = 3

Steven Zheng posted 5 months ago

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