Question

If α and β are the roots of the quadratic equation 3x^2-4x+7=0 find α^3+β^3

Answer 1

We can use Vieta's formulas for quadratic equations. These formulas relate the coefficients of the equation to the sums and products of its roots:

α + β = 4/3 (because α + β = -(-4/3))

αβ = 7/3 (because αβ = 7/3)

Now we want to find α^3 + β^3. This can be factored:

α^3 + β^3 = (α + β)[(α^2 - αβ + β^2)]

We already know α + β = 4/3. Now we'll find the remaining part:

α^2 - αβ + β^2 = (α + β)^2 - 3αβ = (4/3)^2 - 3 * (7/3) = 16/9 - 7 = -1/9

Now we plug this back into the factored form:

α^3 + β^3 = (4/3) * (-1/9) = -4/27