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Let a and b be the roots of the quadratic 2x^2 - 8x + 7 = x^2 - 5x + 3. Compute a^3*b + a*b^3.

Jul 28, 2022

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First, simplify the equation to $$x^2 - 3x + 4 = 0$$

Note that $$a^3b + b^3 a = ab(a^2 + b^2) = ab((a+b)^2 - 2ab)$$

By Vieta's, $$a + b = {b \over a} = -{ -3 \over 1} = 3$$ and $$ab = {c \over a} = 4$$

Can you take it from here?

Jul 28, 2022