Let a and b be the solutions to 5x^2 - 11x + 4 = 4x^2 - 17x + 6. Find 1/a^3 + 1/b^3.
Simplifying the equation, we have x2+6x−2=0
In the form mx^2 + nx + p = 0, the Product of the roots ab = -n/m = -2
So 2ab = -4
Sum of the roots a + b = p/m = -6
Squaring both sides, we have
a2+b2+2ab=36a2+b2−4=36a2+b2=40
1/a3+1/b3=(a3+b3)/(ab)3 Note : a^3 + b^3 = (a + b) (a^2 + b^2 - ab)
So we have
(a+b)(a2+b2−ab)/(ab)3=(−6)(40−−2)/(−8)=(6)(42/8)=31.5
Thus, the answer is 31.5
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