Question

distrubute to simplify the expression (3i-5)[(i+2)+(2i-4)] Fill in the blank
_+_i

Answer 1

That’s not correct

Answer 2

First, we need to simplify the expression \((3i-5)[(i+2)+(2i-4)]\).

1. Start by simplifying the term inside the brackets:
\[
(i+2)+(2i-4) = i + 2 + 2i - 4 = (i + 2i) + (2 - 4) = 3i - 2
\]

2. Now substitute this back into the expression:
\[
(3i-5)(3i-2)
\]

3. Next, distribute the terms:
\[
= 3i \cdot 3i + 3i \cdot (-2) + (-5) \cdot 3i + (-5) \cdot (-2)
\]
\[
= 9i^2 - 6i - 15i + 10
\]
\[
= 9i^2 - 21i + 10
\]

4. Recall that \(i^2 = -1\), so we can replace \(9i^2\) with \(9(-1)\):
\[
= -9 - 21i + 10
\]
\[
= 1 - 21i
\]

Thus, the expression simplifies to \(1 - 21i\).

Now, we can fill in the blank:
\[
1 - 21i \implies \underline{1} + \underline{-21}i
\]

So the answer is \(1 + -21i\).

Answer 3

Let’s recalculate the expression step-by-step to ensure accuracy.

We start with the expression:

\[
(3i-5)\left[(i+2)+(2i-4)\right]
\]

First, simplify the expression inside the brackets:

\[
(i + 2) + (2i - 4) = i + 2 + 2i - 4 = (i + 2i) + (2 - 4) = 3i - 2
\]

Now substitute this back into the original expression:

\[
(3i - 5)(3i - 2)
\]

Now, let's distribute the terms:

1. \(3i \cdot 3i = 9i^2\)
2. \(3i \cdot (-2) = -6i\)
3. \(-5 \cdot 3i = -15i\)
4. \(-5 \cdot (-2) = 10\)

Now, combine these results:

\[
9i^2 - 6i - 15i + 10
\]

Combine the like terms:

\[
9i^2 - 21i + 10
\]

Now, substituting for \(i^2 = -1\):

\[
9(-1) - 21i + 10 = -9 - 21i + 10
\]

Combine the constant terms:

\[
(-9 + 10) - 21i = 1 - 21i
\]

So the simplified expression is:

\[
1 - 21i
\]

In the form requested:

\[
1 + (-21)i
\]

Hence, to fill in the blanks:

\[
\underline{1} + \underline{-21}i
\]

Thank you for your patience, and the result is still \(1 - 21i\). If there are specific parts of this calculation you'd like to revisit or clarify, please let me know!