Consider the following statements in respect of the given equation `(x^(2) + 2)^(2)+ 8x^(2) = 6x (x^(2)+ 2)`

Options

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2

Explanation

To solve the equation \((x^{2} + 2)^{2} + 8x^{2} = 6x (x^{2} + 2)\), we will follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ (x^{2} + 2)^{2} + 8x^{2} = 6x (x^{2} + 2) \]

### Step 2: Substitute \(y = x^{2} + 2\)
Let \(y = x^{2} + 2\). Then the equation becomes:
\[ y^{2} + 8x^{2} = 6xy \]

### Step 3: Rearrange the equation
Rearranging gives us:
\[ y^{2} - 6xy + 8x^{2} = 0 \]

### Step 4: Identify coefficients for the quadratic in \(y\)
This is a quadratic equation in \(y\) where:
- \(a = 1\)
- \(b = -6x\)
- \(c = 8x^{2}\)

### Step 5: Calculate the discriminant
The discriminant \(D\) of the quadratic equation is given by:
\[ D = b^{2} - 4ac \]
Substituting the values:
\[ D = (-6x)^{2} - 4(1)(8x^{2}) = 36x^{2} - 32x^{2} = 4x^{2} \]

### Step 6: Find the roots of the quadratic equation
Using the quadratic formula \(y = \frac{-b \pm \sqrt{D}}{2a}\):
\[ y = \frac{6x \pm \sqrt{4x^{2}}}{2} = \frac{6x \pm 2x}{2} \]
This simplifies to:
\[ y = \frac{8x}{2} = 4x \quad \text{or} \quad y = \frac{4x}{2} = 2x \]

### Step 7: Solve for \(x\) when \(y = 4x\)
Substituting back \(y = x^{2} + 2\):
\[ 4x = x^{2} + 2 \implies x^{2} - 4x + 2 = 0 \]
Calculating the discriminant for this equation:
\[ D = (-4)^{2} - 4(1)(2) = 16 - 8 = 8 \quad (\text{positive, hence real roots}) \]
The sum of the roots is:
\[ \text{Sum} = -\frac{-4}{1} = 4 \]

### Step 8: Solve for \(x\) when \(y = 2x\)
Substituting back \(y = x^{2} + 2\):
\[ 2x = x^{2} + 2 \implies x^{2} - 2x + 2 = 0 \]
Calculating the discriminant for this equation:
\[ D = (-2)^{2} - 4(1)(2) = 4 - 8 = -4 \quad (\text{negative, hence complex roots}) \]
The sum of the roots is:
\[ \text{Sum} = -\frac{-2}{1} = 2 \]

### Step 9: Total roots and their nature
From the two cases:
- From \(y = 4x\), we have 2 real roots.
- From \(y = 2x\), we have 2 complex roots.

Thus, the total number of roots is \(2 + 2 = 4\).

### Summary of Results
- The total number of roots is 4.
- The sum of all roots is \(4 + 2 = 6\).
- All roots are not complex; only 2 are complex.

### Final Statements
- Statement 1: "All the roots of the equation are complex" - **False**
- Statement 2: "The sum of all the roots of the equation is 6" - **True**