How to find the equation of a line, tangent to a circle, that passes through a given external point I am given the equation of a circle: $(x + 2)^2 + (y + 7)^2 = 25$. The radius is $5$. Center of the circle: $(-2, -7)$.

Two lines tangent to this circle pass through point $(4, -3)$, which is outside of said circle. How would I go about finding one of the equations of the lines tangent to the circle?

I haven't started calculus, so I ask for advice fitting for someone starting a topic like this.
A useful technique for this situation which avoids using calculus goes like this:

Find the equation of a line through the point $(4,-3)$ with gradient $k$, and then use this (linear) equation to substitute into the equation of the circle to get a quadratic. The two roots of this quadratic would give the 2 points of intersection of the line with the circle, but in our case, we want to know the values of $k$ for which the quadratic has a double root (i.e. the 2 points of intersection are actually coincident). Use the standard condition ($b^2=4ac$) for a double root of a quadratic.

In your case, the equation of the line through $(4,-3)$ with gradient $k$ is $y+3 = k(x-4)$, or $y+7 = k(x-4)+4$. Substituting this into the equation of the circle gives:

$$(x+2)^2 + (k(x-4)+4))^2 = 25$$

So you need to rearrange this quadratic in $x$ and find out which vales of $k$ give double roots.
Let eq of tangents be y= mx+ c then it passes through point (4,-3) 
So -3=4m+c let it put aside
Now tangent touches circle so radius = distance between centre and line
5 = |-2m+7+c|/(m^2+1)^1/2 
By solving we get
11 m^2 +24m=9
We get two values of m
These are slopes of linesenter preformatted text here
We can equation of lines and angle between lines
This is my idea