Answered

Solve for the expression:

[tex]\((3 - 4i)(3 + 4i)\)[/tex]

A. [tex]\(0 - 7i\)[/tex]
B. [tex]\(25 + 24i\)[/tex]
C. [tex]\(25 - 24i\)[/tex]
D. [tex]\(25 + 0i\)[/tex]



Answer :

To solve the given problem, we need to multiply the two complex numbers [tex]\( (3 - 4i) \)[/tex] and [tex]\( (3 + 4i) \)[/tex].

Recall that when multiplying two complex numbers, we use the distributive property (also known as the FOIL method for binomials):

[tex]\[ (a + bi)(c + di) = ac + adi + bci + bdi^2 \][/tex]

We'll apply this method:

1. [tex]\( 3 \times 3 \)[/tex] which gives [tex]\( 9 \)[/tex].
2. [tex]\( 3 \times 4i \)[/tex] which gives [tex]\( 12i \)[/tex].
3. [tex]\( -4i \times 3 \)[/tex] which gives [tex]\( -12i \)[/tex].
4. [tex]\( -4i \times 4i \)[/tex] which gives [tex]\( -16i^2 \)[/tex].

Combine the real and imaginary parts:

[tex]\[ (3 - 4i)(3 + 4i) = 9 + 12i - 12i - 16i^2 \][/tex]

Notice that the imaginary parts [tex]\( 12i \)[/tex] and [tex]\( -12i \)[/tex] cancel each other out:

[tex]\[ 9 + 12i - 12i - 16i^2 = 9 - 16i^2 \][/tex]

Recall that [tex]\( i^2 = -1 \)[/tex]:

[tex]\[ 9 - 16(-1) = 9 + 16 \][/tex]

Adding these together gives:

[tex]\[ 25 \][/tex]

So, the product of [tex]\( (3 - 4i) \)[/tex] and [tex]\( (3 + 4i) \)[/tex] is [tex]\( 25 + 0i \)[/tex].

The correct answer is:

[tex]\[ 25 + 0i \][/tex]