QuestionEvaluate $(2+4 i)^{2}$ and express the result as $a + b i$ . Simplify your answer completely.

Studdy Solution

STEP 1

Assumptions1. We are given a complex number in the form $a+bi$ where $a=$ and $b=4$ . . We are asked to square this complex number.
3. The result should be in the form $a+bi$ .

STEP 2

We can square a complex number just like we square a binomial. We will use the formula $(a+bi)^2 = a^2 +2abi - b^2$ .
$(2+4i)^2 = (2)^2 +2(2)(4i) - (4i)^2$

STEP 3

Now, calculate the square of the real part, the product of the real and imaginary parts, and the square of the imaginary part.
$(2+i)^2 = +16i -16i^2$

STEP 4

Remember that $i^2 = -1$ . Substitute $-1$ for $i^2$ in the equation.
$(2+4i)^2 =4 +16i -16(-1)$

STEP 5

implify the equation.
$(2+4i)^2 =4 +16i +16$

STEP 6

Combine the real parts and the imaginary parts to write the result in the form $a+bi$ .
$(2+4i)^2 =20 +16i$ So, $(2+4i)^2 =20 +16i$ .

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