How to find the original function, if the tangent is given

Question

So the question goes find $f(x)$ if

$\frac{df}{dx} = 4x - 3$ and the line $y = 5x - 7$ is tangent to $f(x)$.

To find the original function one would integrate the derivative to get $f(x) = 2x^2 - 3x + c$

How do I find $c$ ? And how is the tangent function a part of the question?

Using desmos with trial and error I found c to be 1, but I am unsure how to reach this conclusion mathematically.

Thanks.

Answer 1

The number $c$ must be such that the equation $2x^2-3x+c=5x-7$ has one and only one solution. That is, the equation $2x^2-8x+7+c=0$. That happens if and only if $c=1$.

Answer 2

Slope of straight line is $5$. Set it equal to $4 x -3$. That gives you $x=2$. Corresponding $y$ value is $10-7=3.$ Substitute point coordinates $(2,3)$ into $ y= 2x^2-3x+c $ and that gives you $c=1$.