Question

For which value of c is the line y=3x+c a tangent to the parabola with equation y=x^2-5x+7

Answer 1

Equating the y's:-

3x + c = x^2 - 5x + 7

x^2  - 8x +7 - c = 0

If y = 3x + c is a tangent then there  will be duplicate solutions to this equation 
That is the discriminant b^2 - 4ac = 0. So we have:-

(-8)^2 - 4 * 1 * (7 -c)  = 0
64 -28 + 4c = 0
4c  = - 36

so c = -9  answer