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

_+_i

3 answers

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 \(1 + -21i\).

That’s not correct

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!