To determine which of the given expressions equals [tex]\( x^2 + 25 \)[/tex], we should simplify each expression and compare it to [tex]\( x^2 + 25 \)[/tex].
Option A: [tex]\((x + 5i)^2\)[/tex]
First, let's expand [tex]\( (x + 5i)^2 \)[/tex]:
[tex]\[
(x + 5i)^2 = (x + 5i)(x + 5i)
\][/tex]
Using the distributive property (also known as FOIL method for binomials):
[tex]\[
(x + 5i)(x + 5i) = x^2 + 5ix + 5ix + (5i)^2
\][/tex]
Simplify the terms:
[tex]\[
x^2 + 5ix + 5ix + 25i^2
\][/tex]
Remember that [tex]\( i^2 = -1 \)[/tex]:
[tex]\[
x^2 + 10ix + 25(-1) = x^2 + 10ix - 25
\][/tex]
So,
[tex]\[
(x + 5i)^2 = x^2 + 10ix - 25
\][/tex]
Option B: [tex]\((x + 10i)(x - 15i)\)[/tex]
Next, let's expand [tex]\( (x + 10i)(x - 15i) \)[/tex]:
[tex]\[
(x + 10i)(x - 15i)
\][/tex]
Using the distributive property:
[tex]\[
(x + 10i)(x - 15i) = x^2 - 15ix + 10ix - 150i^2
\][/tex]
Combine like terms and simplify:
[tex]\[
x^2 - 15ix + 10ix - 150(-1) = x^2 - 5ix + 150
\][/tex]
So,
[tex]\[
(x + 10i)(x - 15i) = x^2 - 5ix + 150
\][/tex]
Option C: [tex]\((x - 5i)^2\)[/tex]
Next, let's expand [tex]\( (x - 5i)^2 \)[/tex]:
[tex]\[
(x - 5i)^2 = (x - 5i)(x - 5i)
\][/tex]
Using the distributive property:
[tex]\[
(x - 5i)(x - 5i) = x^2 - 5ix - 5ix + (5i)^2
\][/tex]
Simplify the terms:
[tex]\[
x^2 - 10ix + 25i^2
\][/tex]
Remember that [tex]\( i^2 = -1 \)[/tex]:
[tex]\[
x^2 - 10ix + 25(-1) = x^2 - 10ix - 25
\][/tex]
So,
[tex]\[
(x - 5i)^2 = x^2 - 10ix - 25
\][/tex]
Option D: [tex]\((x - 5i)(x + 5i)\)[/tex]
Finally, let's expand [tex]\( (x - 5i)(x + 5i) \)[/tex]:
[tex]\[
(x - 5i)(x + 5i)
\][/tex]
Using the difference of squares formula:
[tex]\[
(x - 5i)(x + 5i) = x^2 - (5i)^2
\][/tex]
Simplify:
[tex]\[
x^2 - 25i^2
\][/tex]
Remember that [tex]\( i^2 = -1 \)[/tex]:
[tex]\[
x^2 - 25(-1) = x^2 + 25
\][/tex]
So,
[tex]\[
(x - 5i)(x + 5i) = x^2 + 25
\][/tex]
Conclusion
Among all options, only Option D simplifies to [tex]\( x^2 + 25 \)[/tex]. Therefore, the correct answer is:
D. [tex]\((x - 5i)(x + 5i)\)[/tex]
