1. The solutions to x^2 + 5x + 3 = 0 are p and q, and the solutions to x^2 + bx + c = 0 are p^2 and q^2. Find b + c.
2. Let a and b be the solutions to 7x^2 + x - 5 = 0. Compute (a - 4)(b - 4).
Hint for #2:
Factor the equation and let a and b be the roots of it. Then, compute!
1. The solutions to x^2 + 5x + 3 = 0 are p and q, and the solutions to x^2 + bx + c = 0 are p^2 and q^2. Find b + c.
By Vieta
p + q = -5/1 = - 5
Squaring we have that
p^2 + 2pq + q^2 = 25
And
pq = 3/1 = 3 ....so....
2pq = 6
Which implies that
p^2 + 6 + q^2 = 25
p^2 + q^2 = 19
Which means that in x^2 + bx +c
-b/1 = p^2 + q^2
-b = 19
b = -19
And
p^2q^2 = c /1
(pq)^2 = c
(3)^2 = c
9 = c
So
b + c = -19 + 9 = -10