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Mathematics Question on Complex Numbers and Quadratic Equations

If mm is chosen in the quadratic equation (m2+1)x2−3x+(m2+1)2=0(m^2 + 1) x^2 - 3x + (m^2 + 1)^2 = 0 such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is :

A

83 8 \sqrt{3}

B

43 4 \sqrt{3}

C

10510 \sqrt{5}

D

858 \sqrt{5}

Answer

858 \sqrt{5}

Explanation

Solution

SOR=3m2+1⇒(S.O.R)max=3SOR = \frac{3}{m^{2} +1} \Rightarrow \left(S.O.R\right)_{max} = 3
when m=0m = 0
α+β=3\alpha+\beta=3
αβ=1\alpha\beta = 1
∣α2−β2∣=∣∣α−β∣(α2+β2+αβ)∣\left|\alpha^{2} -\beta^{2}\right| =\left| \left|\alpha-\beta\right|\left(\alpha^{2} + \beta^{2} +\alpha\beta\right) \right|
=∣(α−β)2−αβ((α+β)2−αβ)∣= \left|\sqrt{\left(\alpha-\beta\right)^{2}-\alpha\beta} \left(\left(\alpha+\beta\right)^{2} -\alpha\beta\right)\right|
=∣9−4(9−1)∣= \left|\sqrt{9-4} \left(9-1\right)\right|
=5×8= \sqrt{5} \times8