Question

multiply and simplify the following complex numbers:
$(5 + 2i) \cdot (-5 - i)$

Explanation:

Step1: Use the distributive property (FOIL method)

Multiply each term in the first complex number by each term in the second complex number:
$$(5 + 2i) \cdot (-5 - i)=5 \cdot (-5)+5 \cdot (-i)+2i \cdot (-5)+2i \cdot (-i)$$

Step2: Simplify each product

Calculate each product:

• \(5 \cdot (-5)= -25\)
• \(5 \cdot (-i)= -5i\)
• \(2i \cdot (-5)= -10i\)
• \(2i \cdot (-i)= -2i^{2}\)

So the expression becomes:
$$-25 - 5i - 10i - 2i^{2}$$

Step3: Combine like terms and use \(i^{2}=-1\)

Combine the imaginary terms \(-5i - 10i=-15i\), and substitute \(i^{2} = - 1\) into \(-2i^{2}\):
\(-2i^{2}=-2\times(-1) = 2\)
Now the expression is:
$$-25 + 2-15i$$

Step4: Combine the real terms

Combine \(-25\) and \(2\):
$$-23-15i$$

Answer:

\(-23 - 15i\)