+21 Let \(f(x) = x^2 + ax + b\) and \(g(x) = x^2 + cx + d\) be two distinct polynomials with real coefficients such that the x-coordinate of the vertex of f is a root of g and the x-coordinate of the vertex of g is a root of f and both f and g have the same minimum value. If the graphs of the two polynomials intersect at the point (100,-100) what is the value of a+c?
+129899 The x coordinate of the vertex of f = -a/2
The x coordinate of the vertex of g = -c/2
So (-c/2) is a root of f
And (-a/2) is a root of g
So
(-c/2)^2 - a(c/2) + b = 0
(-a/2)^2 - c(a/2) + d = 0 simplify
c^2/4 - ac/2 + b = 0
a^2/4 - ac/2 + d = 0 multiply both equations through by 4
c^2 -2ac + 4b = 0
a^2 - 2ac + 4d = 0 subtract the second equation from the first
c^2 - a^2 + 4 ( b - d) = 0
(c + a) (c - a) = 4 ( d - b) (1)
And since f and g intersect at (100, -100) then
100^2 + 100a + b = 100^2 + 100c + d
100a + b = 100c + d rearrange as
100(a - c) = (d - b) multiply both sides by 4
400 (a - c) = 4 (d - b) (2)
Sub (2) into (1)
(c +a)(c - a) = 400(a - c) divide both sides by ( c - a)
(c + a) = 400( a - c) / ( c - a)
(a + c) = -400 ( c - a) / (c - a)
a + c = -400
