For `i=sqrt(-1)`, which of the following complex number is equivalent to `(10i+4i^2)-(7-3i)`?

Options

A. `-11+7i`
B. `-11+13i`
C. `3-13i`
D. `11-7i`

Answer

To solve the expression \((10i + 4i^2) - (7 - 3i)\), we will follow these steps:

### Step 1: Substitute \(i^2\) with \(-1\)
We know that \(i = \sqrt{-1}\) and thus \(i^2 = -1\). We can substitute \(i^2\) in the expression.

\[ 4i^2 = 4(-1) = -4 \]

### Step 2: Rewrite the expression
Now we can rewrite the expression by substituting \(4i^2\) with \(-4\):

\[ (10i - 4) - (7 - 3i) \]

### Step 3: Distribute the negative sign
Next, we need to distribute the negative sign across the second part of the expression:

\[ 10i - 4 - 7 + 3i \]

### Step 4: Combine like terms
Now, we will combine the like terms. The real parts are \(-4\) and \(-7\), and the imaginary parts are \(10i\) and \(3i\):

1. Combine the real parts:
\[ -4 - 7 = -11 \]

2. Combine the imaginary parts:
\[ 10i + 3i = 13i \]

### Step 5: Write the final answer
Putting it all together, we have:

\[ -11 + 13i \]

Thus, the equivalent complex number is \(-11 + 13i\).

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