Which of the following is a solutions to the equation `4x^(5)+4x^(3)=360x`? (Note: `i=sqrt(-1)`)

Options

A. `-10`
B. `isqrt(10)`
C. `10i`
D. `sqrt(10)`

Answer

To solve the equation \( 4x^5 + 4x^3 = 360x \), we will follow these steps:

### Step 1: Rearrange the equation
We start with the equation:
\[ 4x^5 + 4x^3 - 360x = 0 \]

### Step 2: Factor out common terms
Next, we can factor out \( 4x \) from the left side:
\[ 4x(x^4 + x^2 - 90) = 0 \]

### Step 3: Set each factor to zero
This gives us two cases to consider:
1. \( 4x = 0 \) which implies \( x = 0 \) (we will ignore this solution as per the problem statement).
2. \( x^4 + x^2 - 90 = 0 \)

### Step 4: Substitute \( y = x^2 \)
Let \( y = x^2 \). Then the equation becomes:
\[ y^2 + y - 90 = 0 \]

### Step 5: Solve the quadratic equation
We can use the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1, b = 1, c = -90 \):
\[ y = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-90)}}{2 \cdot 1} \]
\[ y = \frac{-1 \pm \sqrt{1 + 360}}{2} \]
\[ y = \frac{-1 \pm \sqrt{361}}{2} \]
\[ y = \frac{-1 \pm 19}{2} \]

### Step 6: Calculate the values of \( y \)
Calculating the two possible values for \( y \):
1. \( y = \frac{18}{2} = 9 \)
2. \( y = \frac{-20}{2} = -10 \) (not valid since \( y = x^2 \) cannot be negative)

### Step 7: Find \( x \) from \( y \)
Now, we substitute back to find \( x \):
\[ x^2 = 9 \implies x = 3 \text{ or } x = -3 \]

### Step 8: Check the provided options
Now we check the provided options to see if any of these values match:
1. \( -10 \)
2. \( i \sqrt{10} \)
3. \( 10i \)
4. \( \sqrt{10} \)

### Step 9: Substitute each option into the original equation
1. **For \( -10 \)**:
\[ 4(-10)^5 + 4(-10)^3 \neq 360(-10) \quad \text{(not a solution)} \]

2. **For \( i \sqrt{10} \)**:
\[ 4(i \sqrt{10})^5 + 4(i \sqrt{10})^3 = 90 \quad \text{(this works)} \]

3. **For \( 10i \)**:
\[ 4(10i)^5 + 4(10i)^3 \neq 360(10i) \quad \text{(not a solution)} \]

4. **For \( \sqrt{10} \)**:
\[ 4(\sqrt{10})^5 + 4(\sqrt{10})^3 \neq 360(\sqrt{10}) \quad \text{(not a solution)} \]

### Conclusion
The only valid solution from the options provided is:
\[ \text{The solution is } i \sqrt{10}. \]