To find the complex number [tex]\(x\)[/tex] such that the product of [tex]\( (3 - 4i) \)[/tex] and [tex]\( x \)[/tex] is 25, use the following steps:
1. Set up the equation:
[tex]\[
(3 - 4i) \cdot x = 25
\][/tex]
2. Isolate [tex]\( x \)[/tex]:
To solve for [tex]\( x \)[/tex], divide both sides of the equation by [tex]\( (3 - 4i) \)[/tex]:
[tex]\[
x = \frac{25}{3 - 4i}
\][/tex]
3. Multiply by the conjugate:
To simplify the complex division, multiply the numerator and the denominator by the conjugate of the denominator, [tex]\( 3 + 4i \)[/tex]:
[tex]\[
x = \frac{25 \cdot (3 + 4i)}{(3 - 4i)(3 + 4i)}
\][/tex]
4. Simplify the denominator:
The product of a complex number and its conjugate is a real number:
[tex]\[
(3 - 4i)(3 + 4i) = 3^2 - (4i)^2 = 9 - (-16) = 9 + 16 = 25
\][/tex]
So the equation now looks like:
[tex]\[
x = \frac{25 \cdot (3 + 4i)}{25}
\][/tex]
5. Simplify the fraction:
Divide both the numerator and the denominator by 25:
[tex]\[
x = 3 + 4i
\][/tex]
Thus, the complex number [tex]\( x \)[/tex] that, when multiplied by [tex]\( (3 - 4i) \)[/tex], yields 25 is [tex]\( 3 + 4i \)[/tex].
So, the correct answer is:
[tex]\[ 3 + 4i \][/tex]
