Find C so that the line y = 4x + 3 is tangent to the curve y = x^2 + C.

At the point of tangency:
${y'}_{\text{line}} = {y'}_{\text{parabola}}$

$4 = 2x$

$x = 2$
 

At x = 2
$y = 4(2) + 3 = 11$
 

At the point of tangency (2, 11):
$11 = 2^2 + C$

$C = 7$