If $m$ is chosen in the quadratic equation $\left(m^2+1\right) x^2-3 x+\left(m^2+1\right)^2=0$ such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is:
Select the correct option:
A
$8 \sqrt{3}$
B
$10 \sqrt{5}$
C
$4 \sqrt{3}$
D
$8 \sqrt{5}$
✓ Correct! Well done.
✗ Incorrect. Try again or view the solution.
Solution
Sum of roots
For maximum $\mathrm{m}=0$
Hence equation becomes $x^2-3 x+1=0$
$$
\left|\alpha^3-\beta^3\right|=\left|(\alpha-\beta)\left(\alpha^2+\beta^2+\alpha \beta\right)\right|=\sqrt{5}(9-1)=8 \sqrt{5}
$$
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