A sector occupies the same fraction of the circle as its arc occupies of the circumference. This leads to $\text{sector area}=\frac12r(\text{arc length})$.

Your equation in $r$ may have two solutions. Remember that a circle's radius cannot be negative.

Algebra is the fastest method here: use the arc information to obtain $r^2-7r-98=0$.

Graph $y=x^2-7x-98$. The positive $x$-intercept is the possible radius; disregard the negative intercept because a radius must be positive.

Let $s$ be the length of the minor arc. The major and minor arcs together make the full circumference, so their total length is $2\pi r$.

Because the major arc is $14\pi$ units longer than the minor arc, the minor arc is

The minor sector has the same fraction of the circle's area as its arc has of the circumference. Therefore,

Substitute the given area and the expression for $s$:

Divide by $\pi$ and multiply by $2$:

Thus,

Factoring gives $\left(r-14\right)\left(r+7\right)=0$. A radius must be positive, so the radius is $14$.