Let's solve the problems step-by-step. I'll start with the first set of problems, which are solved using the square root property.

Square Root Property:

13. $x^2 + 100 = 0$

1. Subtract 100 from both sides: $x^2 = -100$
2. Take the square root of both sides: $x = \pm \sqrt{-100}$
3. Simplify: $x = \pm 10i$ Solution: $x = 10i, -10i$

14. $2x^2 + 27 = -13$

1. Subtract 27 from both sides: $2x^2 = -40$
2. Divide by 2: $x^2 = -20$
3. Take the square root of both sides: $x = \pm \sqrt{-20}$
4. Simplify: $x = \pm \sqrt{20}i = \pm 2\sqrt{5}i$ Solution: $x = 2\sqrt{5}i, -2\sqrt{5}i$

Solving by Completing the Square:

15. $x^2 + 2x + 1 = 16$

1. Recognize this is already a perfect square trinomial: $(x + 1)^2 = 16$
2. Take the square root: $x + 1 = \pm 4$
3. Solve for $x$: $x = -1 \pm 4$ $x = 3 \text{ or } x = -5$ Solution: $x = 3, -5$

16. $x^2 - 10x + 25 = 3$

1. Subtract 3 from both sides to complete the square: $(x - 5)^2 = 3$
2. Take the square root: $x - 5 = \pm \sqrt{3}$
3. Solve for $x$: $x = 5 \pm \sqrt{3}$ Solution: $x = 5 + \sqrt{3}, 5 - \sqrt{3}$

Let me know if you'd like the next set of problems solved or further details about any solution. Here's a related expansion to help:

1. What is the square root property, and how is it used for equations involving imaginary numbers?
2. How can you identify perfect square trinomials for easier solving?
3. What is the significance of simplifying roots, particularly with $i$ (imaginary)?
4. Can every quadratic equation be solved by completing the square? Why or why not?
5. How does the discriminant ($b^2 - 4ac$) relate to the type of solutions?

Tip: When using the square root property, always simplify the square root first before adding or subtracting constants.