Sum and product of roots?

Question

Given that α and β are the roots of the equation x^2 = 3x + 5,
show that α^4 = 57α + 70.

Sum of roots --> 3
Product of roots --> -5

Sum of roots --> α + β = 3
Product of roots --> αβ = -5

Note: The roots of the equation are not intergers. As a result, you cannot say that the sum cannot be 3 while the product is 5.

The sum and product of roots are calculated by the formulas -b/a and c/a respectively where the equation is
ax^2 + bx + c = 0

Answer 1

If α is a root of the equation, then we have:

α² = 3α + 5

Squaring both sides:

α⁴ = 9α² + 30α + 25

Of course, α² = 3α+5, so substituting:

α⁴ = 9(3α + 5) + 30α + 25
α⁴ = 27α + 45 + 30α + 25
α⁴ = 57α + 70

Q.E.D.

Answer 2

x^2 = 3x + 5
x^2-3x-5=0
α+β =3
α β =-5

β=3-α
α (3-α )=-5
3α-α^2=-5
α^2=3α+5

(α^2)^2=(3α+5)^2
α^4 =9 α^2+30 α+25
=9(3α+5)+30α+25
=27α+45+30α+25
=57α+70#