The values of 'K' for which the equation `|x|^2 (|x|^2 – 2k + 1) = 1 - k^2`, has No root, when k belongs to

Options

A. `( -oo,-1)`
B. (-1,1)
C. `(1,5/4)`
D. R

Answer

To solve the equation \( |x|^2 (|x|^2 - 2k + 1) = 1 - k^2 \) for the values of \( k \) for which it has no roots, we can follow these steps:

### Step 1: Rewrite the Equation
We start with the given equation:
\[ |x|^2 (|x|^2 - 2k + 1) = 1 - k^2 \]
Let \( y = |x|^2 \). Then, we can rewrite the equation as:
\[ y(y - 2k + 1) = 1 - k^2 \]

### Step 2: Expand and Rearrange
Expanding the left-hand side gives us:
\[ y^2 - (2k - 1)y - (1 - k^2) = 0 \]
This is a quadratic equation in \( y \):
\[ y^2 - (2k - 1)y + (k^2 - 1) = 0 \]

### Step 3: Determine Conditions for No Roots
For this quadratic equation to have no real roots, the discriminant must be less than zero. The discriminant \( D \) of a quadratic equation \( ay^2 + by + c = 0 \) is given by:
\[ D = b^2 - 4ac \]
In our case:
- \( a = 1 \)
- \( b = -(2k - 1) \)
- \( c = k^2 - 1 \)

Thus, the discriminant is:
\[ D = (-(2k - 1))^2 - 4(1)(k^2 - 1) \]
\[ D = (2k - 1)^2 - 4(k^2 - 1) \]

### Step 4: Simplify the Discriminant
Now we simplify the discriminant:
\[ D = (2k - 1)^2 - 4(k^2 - 1) \]
Expanding both terms:
\[ D = (4k^2 - 4k + 1) - (4k^2 - 4) \]
\[ D = 4k^2 - 4k + 1 - 4k^2 + 4 \]
\[ D = -4k + 5 \]

### Step 5: Set the Discriminant Less Than Zero
To find the values of \( k \) for which there are no roots, we set the discriminant less than zero:
\[ -4k + 5 < 0 \]
Solving for \( k \):
\[ 5 < 4k \]
\[ \frac{5}{4} < k \]
Thus, the values of \( k \) for which the equation has no roots are:
\[ k > \frac{5}{4} \]

### Final Answer
The values of \( k \) for which the equation \( |x|^2 (|x|^2 - 2k + 1) = 1 - k^2 \) has no roots is:
\[ k \in \left( \frac{5}{4}, \infty \right) \]