A parabola with equation y = x^2 + bx + c passes through the points (2,3) and (4,3). What is c?

Instead of knowing the constants and solving for a point, we are given two points and asked to solve for the constants.

First plug in the two points to the equation.

 

(2,3)=(x,y) plugged into y=x^2+bx+c

3=2^2+2b+c

3=4+2b+c

subtract 4 from each side

-1=2b+c

 

(4,3)

3=4^2+4b+c

3=16+4b+c

subtract 16 from each side

-13=4b+c

 

Now this problem is only a matter of solving a system of equations:

-13=4b+c

-1=2b+c

 

Subtract the second one from the first one to eliminate c:

-12=2b

divide by 2

b= -6

 

Substitute -6 in for b to get c:

-1=2(-6)+c

-1=-12+c

add 12 to each side

11=c