Multiply. $$ [x-(3+2 i)][x-(3-2 i)] $$ Note that these expressions contain complex numbers. Simplify your answer as much as possible. $$ \square $$

Question

UpStudy AI · Accepted Answer

We start with the expression:
$$
\left[x - (3+2i)\right]\left[x - (3-2i)\right]
$$
Notice that we have two factors of the form $$\left(x - 3 \mp 2i\right)$$.

Step 1. Write the expression in a suggestive form:
$$
\left(x-3-2i\right)\left(x-3+2i\right)
$$

Step 2. Recognize that this product is in the form of a difference of squares:
$$
(a - b)(a + b) = a^2 - b^2
$$
where in our case, $$a = x-3$$ and $$b = 2i$$.

Step 3. Apply the formula:
$$
\left(x-3\right)^2 - \left(2i\right)^2
$$

Step 4. Simplify $$\left(2i\right)^2$$:
$$
\left(2i\right)^2 = 4i^2
$$
and since $$i^2 = -1$$, we have:
$$
4i^2 = 4(-1) = -4
$$

Step 5. Substitute back into the expression:
$$
\left(x-3\right)^2 - (-4) = \left(x-3\right)^2 + 4
$$

Step 6. (Optional) Expand $$\left(x-3\right)^2$$ if desired:
$$
\left(x-3\right)^2 = x^2 - 6x + 9
$$
so the expression becomes:
$$
x^2 - 6x + 9 + 4 = x^2 - 6x + 13
$$

Thus, the simplified form is:
$$
\boxed{x^2 - 6x + 13}
$$
The simplified form of the expression is $$ x^2 - 6x + 13 $$.

Multiply.

Note that these expressions contain complex numbers.
Simplify your answer as much as possible.

Solución de inteligencia artificial de Upstudy

Responder

Solución

The Deep Dive

preguntas relacionadas

Latest Algebra Questions

Multiply. Note that these expressions contain complex numbers. Simplify your answer as much as possible.