If 2x `=3+5i`, then what is the `2x^(3) + 2x^(2) - 7x + 72` ?

Options

A. 4
B. `-4`
C. 8
D. `-8`

Answer

To solve the problem, we start with the equation given:

**Given:**
\[ 2x = 3 + 5i \]

We need to find the value of the expression:
\[ 2x^3 + 2x^2 - 7x + 72 \]

### Step 1: Solve for \( x \)
First, we can isolate \( x \) from the equation \( 2x = 3 + 5i \).

\[ x = \frac{3 + 5i}{2} \]

### Step 2: Calculate \( x^2 \)
Next, we calculate \( x^2 \).

\[ x^2 = \left(\frac{3 + 5i}{2}\right)^2 = \frac{(3 + 5i)(3 + 5i)}{4} = \frac{9 + 30i - 25}{4} = \frac{-16 + 30i}{4} = -4 + \frac{15}{2}i \]

### Step 3: Calculate \( x^3 \)
Now we calculate \( x^3 \) using the value of \( x \).

\[ x^3 = x \cdot x^2 = \left(\frac{3 + 5i}{2}\right)\left(-4 + \frac{15}{2}i\right) \]

Calculating this product:

\[ = \frac{(3 + 5i)(-4 + \frac{15}{2}i)}{2} = \frac{-12 + \frac{45}{2}i - 20i - \frac{75}{2}}{2} \]
\[ = \frac{-12 - \frac{75}{2} + \frac{45}{2}i - 20i}{2} \]
\[ = \frac{-\frac{24}{2} - \frac{75}{2} + \frac{45}{2}i - \frac{40}{2}i}{2} \]
\[ = \frac{-\frac{99}{2} - \frac{-5}{2}i}{2} = -\frac{99}{4} - \frac{5}{4}i \]

### Step 4: Substitute \( x, x^2, x^3 \) into the expression
Now we substitute \( x, x^2, x^3 \) into the expression \( 2x^3 + 2x^2 - 7x + 72 \).

\[ 2x^3 = 2\left(-\frac{99}{4} - \frac{5}{4}i\right) = -\frac{198}{4} - \frac{10}{4}i = -49.5 - 2.5i \]
\[ 2x^2 = 2\left(-4 + \frac{15}{2}i\right) = -8 + 15i \]
\[ -7x = -7\left(\frac{3 + 5i}{2}\right) = -\frac{21}{2} - \frac{35}{2}i \]

Now, we combine everything:

\[ 2x^3 + 2x^2 - 7x + 72 = \left(-49.5 - 2.5i - 8 + 15i - \frac{21}{2} - \frac{35}{2}i + 72\right) \]

### Step 5: Combine like terms
Combining the real parts and the imaginary parts:

Real part:
\[ -49.5 - 8 + 72 - \frac{21}{2} = -49.5 - 8 + 72 - 10.5 = 4 \]

Imaginary part:
\[ -2.5i + 15i - \frac{35}{2}i = -2.5i + 15i - 17.5i = -5i \]

### Final Result
Thus, the final result is:

\[ 2x^3 + 2x^2 - 7x + 72 = 4 - 5i \]