Question

Let α and β be the roots of the equation ax² + 2bx + c = 0 and α  + k, β + k be the roots of Ax² + 2Bx + C = 0. Then prove that A² (b² − ac) = a²(B² − AC).

Answer 1

Let

f(x) = x² + 2bx/a + c/a and g(x) = x² + 2Bx/A + C/A

As roots of f(x) = 0 are  α and β  and roots of g(x) = 0 are α  + k and β + k, graph of g(x) will be obtained by translating the graph of f(x) by k units on the x-axis Fig.

Hence, Minimum value of f(x) = Minimum value of g(x).

⇒ A²(b² - ac) = (B² - AC)a²