1. If a + b = 5 and ab = 3, compute the value of a^3 + b^3 2. Solve for all values of x: (x-3)^2 -5(x-3)+6=0

Question

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Answer 1

a³ + b³ = (a + b)(a² - ab + b²) = 5(a² + b² - 3)

(a + b)² = a² + 2ab + b². So, 25 = a² + b² + 6. Therefore, a² + b² = 19.

So, a³ + b³ = 5(a² + b² - 3) = 5(19 - 3) = 80

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(x-3)² - 5(x-3) + 6 = 0

Let u = x-3. Then, u² - 5u + 6 = 0

(u - 3)(u - 2) = 0. So, u = 3 or u = 2.

That is, x - 3 = 3 or 2.

So, x = 6 or 5.