TITLE: Find the exact values of m.
QUESTION [6 upvotes]: I'm in an American high school but my teacher gave me some IBDP questions and I'm struggling to solve one of them:
Find the exact values of $m$ for which the line $y= mx+3$ is tangent to the curve with equation $y=3x^2 -x +5$

REPLY [3 votes]: We need to find $x$ values for which the slopes are the same, and the $y$ values are the same. That's what tangent means. Let $f(x) = mx + 3$ and let $g(x) = 3x^2 - x + 5$.
First take the derivative of $g(x)$, which is
$$
g'(x) = 6x - 1.
$$
This will give you the slope at any given point.
We want $m$ to be equal to $g'(x)$, so that they have the same slope. We also want $f(x)$ to equal $g(x)$. So we have the two equations:
$$
m = 6x - 1
$$
and
$$
mx + 3 = 3x^2 - x + 5.
$$
We can substitute $6x - 1$ for $m$ in the second equation
$$
(6x - 1)x + 3 = 3x^2 - x + 5
$$
and solve for $x$.