+1418 Let a and b be the roots of the quadratic x^2 - 5x + 3 = 2x^2 + 15x - 10. Find the quadratic whose roots are a^2 + a + 1 and b^2 + b + 1.
+129771 Simplify as
x^2 + 20x - 13 = 0
Product of the roots = ab = -13 (ab)^2 = 169
2ab = -26
Sum of the roots = a + b = -20
(a + b)^2 = (-20)^2
a^2 + 2ab + b^2 = 400
a^2 + b^2 + 2ab = 400
a^2 + b^2 - 26 = 400
a^2 + b^2 = 426
For the new quadratic, the sum of the roots = (a^2 + b^2) + (a + b) + 2 = 426 - 20 + 2 = 408
The product of the roots = (a^2 + a + 1) (b^2 + b + 1) =
a^2 b^2 + a^2 b + a^2 + a b^2 + a b + a + b^2 + b + 1 =
(a^2 + b^2) + (ab)^2 + ab (a + b) + ab + (a + b) + 1 =
426 + 169 + (-13)(-20) -13 - 20 + 1 = 823
The quadratic is Ax^2 + Bx + C and we can let A = 1
x^2 - 408x + 823
