The values of x in the equation `a^(2)b^(2)x^(2)-(a^(2)+b^(2))x+1,ane1,bne0` is

Options

A. `(1)/(a^(2))`
B. `(1)/(b^(2))`
C. `(1)/(a^(2)),(1)/(b^(2))`
D. None of these

Answer

To solve the quadratic equation \( a^2b^2x^2 - (a^2 + b^2)x + 1 = 0 \), we can follow these steps:

### Step 1: Identify the coefficients
The given equation is in the standard form of a quadratic equation \( Ax^2 + Bx + C = 0 \), where:
- \( A = a^2b^2 \)
- \( B = -(a^2 + b^2) \)
- \( C = 1 \)

### Step 2: Use the quadratic formula
The solutions for \( x \) can be found using the quadratic formula:
\[ x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]
Substituting the values of \( A \), \( B \), and \( C \):
\[ x = \frac{-(- (a^2 + b^2)) \pm \sqrt{(-(a^2 + b^2))^2 - 4(a^2b^2)(1)}}{2(a^2b^2)} \]

### Step 3: Simplify the expression
This simplifies to:
\[ x = \frac{(a^2 + b^2) \pm \sqrt{(a^2 + b^2)^2 - 4a^2b^2}}{2a^2b^2} \]

### Step 4: Further simplification of the discriminant
Now, we simplify the discriminant:
\[ (a^2 + b^2)^2 - 4a^2b^2 = a^4 + 2a^2b^2 + b^4 - 4a^2b^2 = a^4 - 2a^2b^2 + b^4 = (a^2 - b^2)^2 \]
Thus, the expression for \( x \) becomes:
\[ x = \frac{(a^2 + b^2) \pm (a^2 - b^2)}{2a^2b^2} \]

### Step 5: Calculate the two possible values for \( x \)
This gives us two cases:
1. \( x_1 = \frac{(a^2 + b^2) + (a^2 - b^2)}{2a^2b^2} = \frac{2a^2}{2a^2b^2} = \frac{1}{b^2} \)
2. \( x_2 = \frac{(a^2 + b^2) - (a^2 - b^2)}{2a^2b^2} = \frac{2b^2}{2a^2b^2} = \frac{1}{a^2} \)

### Step 6: Final values of \( x \)
Thus, the values of \( x \) are:
\[ x = \frac{1}{b^2} \quad \text{and} \quad x = \frac{1}{a^2} \]

### Summary of the solution:
The values of \( x \) in the equation \( a^2b^2x^2 - (a^2 + b^2)x + 1 = 0 \) are:
\[ x = \frac{1}{a^2} \quad \text{and} \quad x = \frac{1}{b^2} \]