Let be an ordered pair of real numbers that satisfies the equation x^2+y^2=14x+48y . What is the minimum value of x?

This is going to be a circe, just did a similar one for you.

Once you have the centre of the circle the minimim x value will be the centre x value minus the radius.

x^2 -14x + y^2 - 48y = 0

x^2 - 14x + 49 + y^2 - 48y + 576 = 49 + 576

(x - 7)^2 + ( y - 24)^2 = 625

x is minimized/maximized when y = 24

(x - 7)^2 = 625 take both roots

x - 7 = 25 or x - 7 = -25

x = 32 or x = - 18

So....x = -18 is the minimum value for x