Theory question 110
Polynomials(a) If α and β are the roots of the equation 2x2 + 5x − 6 = 0, find the equation whose roots are (α − 2) and (β − 2).
(b) Given that ∫0k(x2 − 2x)dx = 4/3, find the values of k.
Show the model answer
Model answer
(a) α + β = −5/2, αβ = −3
Sum of new roots = α + β − 4 = −13/2
Product = αβ − 2(α + β) + 4 = −3 + 5 + 4 = 6
x2 + (13/2)x + 6 = 0, i.e. 2x2 + 13x + 12 = 0
(b) [x3/3 − x2] from 0 to k = k3/3 − k2 = 4/3
k3 − 3k2 − 4 = 0
This cubic has only one real root: k ≈ 3.36 (by trial or iteration)
Explanation
Find the sum and product of the new roots from those of the old ones; for (b) integrate, substitute the limits and solve for k.