Which is the value of the sum x1^3 + x2^3 if x1,x2 are solutions of the equation x^2-5x+6=0

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f(x) = x^2 - 5x + 6

x1 and x2 are solution

We need to determine the value of x1^3 + x2^3

From viete's rule, we know that:

x1+ x2 = -b/a = 5.........(1)

x1*x2 = c/a = 6...........(2)

But f(x1) = f(x2) =0

==> f(x1) = x1^2 -5x1 +...

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f(x) = x^2 - 5x + 6

x1 and x2 are solution

We need to determine the value of x1^3 + x2^3

From viete's rule, we know that:

x1+ x2 = -b/a = 5.........(1)

x1*x2 = c/a = 6...........(2)

But f(x1) = f(x2) =0

==> f(x1) = x1^2 -5x1 + 6 = 0 ......(3)

==> f(x2) = x2^2 - 5x2 + 6 =0 ...(4)

Multiply 3 by x1 and (4) by x2

==> x1^3 - 5x1^2 + 6x1 = 0

==> x2^3 - 5x2^2 + 6x2 = 0

Add both equations:

==> x1^3 + x2^3 = 5x1^2 + 5x2^2 - 6x1 -6x2

==> x1^3 + x2^3 = 5(x1^2 + x2^2) - 6(x1+x2)

                             = 5[(x1+x2)^2 - 2x1*x2] - 6(x1+x2)

                            = 5(5^2) - 5(2*6) - 6(5)

                            = 125 - 60 - 30

                            =125 - 90 = 35

==> x1^3 + x2^3 = 35

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