Question

select the correct answer.
what is the value of this expression?
(5 + 3i) − (5 + 3i)(5 − 5i)
a. 35 − 13i
b. -5 − 37i
c. -5 + 37i
d. -35 + 13i

Explanation:

Step1: Factor out \((5 + 3i)\)

We can factor out \((5 + 3i)\) from the expression \((5 + 3i)-(5 + 3i)(5 - 5i)\) using the distributive property \(ab - ac=a(b - c)\). So we get \((5 + 3i)[1-(5 - 5i)]\).

Step2: Simplify the bracket

Simplify the expression inside the bracket: \(1-(5 - 5i)=1 - 5+5i=-4 + 5i\). Now our expression becomes \((5 + 3i)(-4 + 5i)\).

Step3: Multiply the complex numbers

Use the FOIL method to multiply \((5 + 3i)(-4 + 5i)\).

• First terms: \(5\times(-4)=-20\)
• Outer terms: \(5\times5i = 25i\)
• Inner terms: \(3i\times(-4)=-12i\)
• Last terms: \(3i\times5i=15i^{2}\)

Since \(i^{2}=-1\), the last term becomes \(15\times(-1)=-15\). Now combine all the terms: \(-20+25i-12i - 15\).

Step4: Combine like terms

Combine the real parts and the imaginary parts separately.

• Real parts: \(-20-15=-35\)
• Imaginary parts: \(25i-12i = 13i\)

So the result is \(-35 + 13i\).

Answer:

D. \(-35 + 13i\)