Find point c on the x-axis so ac+bc is a minimum with A(-3,2) and B(-6,-4)

It's on the x-axis so y = 0

A (-3 , 2)
B (-6 , -4)
C (x , 0)

AC =
sqrt((-3 - x)^2 + (2 - 0)^2) =
sqrt(9 + 6x + x^2 + 4) =
sqrt(x^2 + 6x + 13)

BC =
sqrt((-6 - x)^2 + (-4)^2) =>
sqrt(36 + 12x + x^2 + 16) =>
sqrt(x^2 + 12x + 52)

sqrt(x^2 + 6x + 13) + sqrt(x^2 + 12x + 52) = D

find when dD/dx = 0

(2x + 6) * (x^2 + 6x + 13)^(-1/2) + (2x + 12) * (x^2 + 12x + 52)^(-1/2) = 0
((2x + 6) * sqrt(x^2 + 12x + 52) + (2x + 12) * sqrt(x^2 + 6x + 13)) / sqrt((x^2 + 6x + 13) * (x^2 + 12x + 52)) = 0
(2x + 6) * sqrt(x^2 + 12x + 52) + (2x + 12) * sqrt(x^2 + 6x + 13) = 0
(2x + 6) * sqrt(x^2 + 12x + 52) = -(2x + 12) * sqrt(x^2 + 6x + 13)
(x + 3) * sqrt(x^2 + 12x + 52) = -(x + 6) * sqrt(x^2 + 6x + 13)
(x + 3)^2 * (x^2 + 12x + 52) = (x + 6)^2 * (x^2 + 6x + 13)
(x^2 + 6x + 9) * (x^2 + 12x + 52) = (x^2 + 12x + 36) * (x^2 + 6x + 13)
x^4 + 12x^3 + 52x^2 + 6x^3 + 72x^2 + 312x + 9x^2 + 108x + 468 = x^4 + 6x^3 + 13x^2 + 12x^3 + 72x^2 + 156x + 36x^2 + 216x + 468
x^4 + 18x^3 + 133x^2 + 420x + 468 = x^4 + 18x^3 + 121x^2 + 372x + 468
133x^2 + 420x = 121x^2 + 372x
12x^2 + 48x = 0
x * (x + 4) = 0
x = 0 , -4

sqrt(x^2 + 6x + 13) + sqrt(x^2 + 12x + 52) = D
x = 0

sqrt(0 + 0 + 13) + sqrt(0 + 0 + 52) =>
sqrt(13) + 2 * sqrt(13) =>
3 * sqrt(13)

x = -4
sqrt(16 - 24 + 13) + sqrt(16 - 48 + 52) =>
sqrt(5) + sqrt(20) =>
sqrt(5) + 2 * sqrt(5) =>
3 * sqrt(5)

The minimum distance occurs when C is (-4 , 0)