Multiply.
x-(6+2i)][x-(6-2i)]
Note that these expressions contain complex numbers.
Simplify your answer as much as possible.
Let a=6+2i, b=6-2i
(x-a)(x-b)
=x²-(a+b)x+ab
a+b = 12
ab=6²-(2i)²
=36-(-4)
=40
So
(x-a)(x-b)
=x²-12x+40
To multiply the given expressions, we can use the FOIL method (First, Outer, Inner, Last).
First, let's expand the expression (x - (6 + 2i))(x - (6 - 2i)) using FOIL:
= x(x) - x(6 - 2i) - (6 + 2i)(x) + (6 + 2i)(6 - 2i)
= x^2 - 6x + 2ix - 6x + 12i - 2i^2 + 6x - 12i + 2i^2
Note that i^2 = -1, so the terms 2ix and -2i^2 will cancel each other out.
= x^2 - 6x + 6x + 12i - 12i
= x^2 + 0 + 0i
The simplified answer to the multiplication is x^2.
To multiply the given expressions, we can use the binomial multiplication formula.
The formula for multiplying two binomials is:
(a + b)(c + d) = ac + ad + bc + bd
Let's apply this formula to the given expressions:
(x - (6 + 2i))(x - (6 - 2i))
Step 1: Distribute the x term to both expressions.
x * x - x * (6 - 2i) - (6 + 2i) * x + (6 + 2i) * (6 - 2i)
Step 2: Simplify each term.
x^2 - x * 6 + x * 2i - (6 + 2i) * x + 6 * (6 - 2i) + 2i * (6 - 2i)
Step 3: Apply the multiplication of terms.
x^2 - 6x + 2ix - (6x - 2ix + 36 - 12i + 12i - 4i^2)
Step 4: Simplify further using the fact that i^2 is equal to -1.
x^2 - 6x + 2ix - (6x - 2ix + 36 - 12i + 12i + 4)
Step 5: Combine like terms.
x^2 - 6x + 2ix - 6x + 2ix + 36 - 12i + 12i + 4
Step 6: Simplify further.
x^2 - 12x + 40
Therefore, the simplified expression for (x - (6 + 2i))(x - (6 - 2i)) is x^2 - 12x + 40.