Question

simplify.
(3 - i)(3 + i)
write your answer in the form a + bi.

Explanation:

Step1: Apply the difference of squares formula

The expression \((3 - i)(3 + i)\) is in the form of \((a - b)(a + b)\), which can be simplified using the difference of squares formula \((a - b)(a + b)=a^2 - b^2\). Here, \(a = 3\) and \(b = i\). So we have:
\((3 - i)(3 + i)=3^2 - i^2\)

Step2: Simplify the squares

We know that \(3^2 = 9\) and \(i^2=- 1\) (by the definition of the imaginary unit \(i\)). Substituting these values into the expression:
\(3^2 - i^2=9-(-1)\)

Step3: Simplify the subtraction of a negative number

Simplify \(9-(-1)\). Subtracting a negative number is the same as adding its positive counterpart, so \(9 - (-1)=9 + 1\)

Step4: Calculate the sum

\(9+1 = 10\). We can write \(10\) in the form \(a+bi\) as \(10+0i\) where \(a = 10\) and \(b = 0\)

Answer:

\(10 + 0i\) (or simply \(10\))