1. **State the problem:** We need to find the equation of a circle centered at point $B(1,4)$ with radius $5$. 2. **Formula for a circle:** The equation of a circle with center $(h,k)$ and radius $r$ is $$ (x - h)^2 + (y - k)^2 = r^2 $$ 3. **Substitute the given values:** Here, $h=1$, $k=4$, and $r=5$. So, $$ (x - 1)^2 + (y - 4)^2 = 5^2 $$ 4. **Simplify the radius squared:** $$ (x - 1)^2 + (y - 4)^2 = 25 $$ 5. **Check the x-intercepts:** When $y=0$, $$ (x - 1)^2 + (0 - 4)^2 = 25 $$ $$ (x - 1)^2 + 16 = 25 $$ $$ (x - 1)^2 = 9 $$ $$ x - 1 = \pm 3 $$ $$ x = 1 \pm 3 $$ So the x-intercepts are at $x = -2$ and $x = 4$, which matches the problem statement. **Final answer:** $$ (x - 1)^2 + (y - 4)^2 = 25 $$