How to solve when the unknown is given?

Question

A curve has a gradient function $px^2 - 5x$, where $p$ is a constant . The tangent to the curve at the point $x=1$ is parallel to the straight line $y+2x-5=0$. Find the value of $p$.

Answer 1

The straight line $y+2x-5=0$ has slope $-2$.

The slope of the curve at $x=1$ is $p(1)^2-5(1)=p-5$.

For the curve to be parallel to the line at the point, the slopes must be equal, so:

$p-5=-2$, which gives $p=3$.

Answer 2

The curve's gardient is $y' = f'(x) = p x^2 - 5x$. The straight line is $y + 2x - 5 = 0$ which is $y = -2x + 5$. We know that the slope of $y$ at $x=1$, that is $f'(1)$, is equal to $-2$, the slope of the straight line. Therefore $$ -2 = f'(1) = p - 5 $$ which yields $p = 3$.