Question

With what integer a is the the graph `y=4x-2` the tangent line of `y=ax^2+28x+14`?

Answer 1

`a = 9`

Explanation:

The given equations will have the same y value at point of tangency, therefore, we may set the right sides of both equation equal:

`4x-2= ax^2+28x+14`

`0 = ax^2+24x+16" [1]"`

The slope of the tangent is 4, therefore, we must set the first derivative of the quadratic equal to 4:

`dy/dx = 2ax+28`

`4 = 2ax+28`

`-24=2ax`

`x = -12/a" [2]"`

The substitute equation [2] into equation [1]:

`0 = a(-12/a)^2+24(-12/a)+16`

`0 = 144/a-288/a+16`

`0 = 144-288+16a`

`16a = 144`

`a = 9`