Solve the following quations by using qardratic formula:
`p^(2)x^(2)+(p^(2)-q^(2))x-q^(2)=0`

Answer

To solve the quadratic equation \( p^2 x^2 + (p^2 - q^2)x - q^2 = 0 \) using the quadratic formula, we follow these steps:

### Step 1: Identify the coefficients
The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \). Here, we can identify:
- \( a = p^2 \)
- \( b = p^2 - q^2 \)
- \( c = -q^2 \)

### Step 2: Calculate the discriminant
The discriminant \( D \) is given by the formula:
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = (p^2 - q^2)^2 - 4(p^2)(-q^2) \]
Calculating \( D \):
\[ D = (p^2 - q^2)^2 + 4p^2q^2 \]
Expanding \( (p^2 - q^2)^2 \):
\[ D = p^4 - 2p^2q^2 + q^4 + 4p^2q^2 \]
Combining like terms:
\[ D = p^4 + 2p^2q^2 + q^4 \]
This can be factored as:
\[ D = (p^2 + q^2)^2 \]

### Step 3: Apply the quadratic formula
The quadratic formula for the roots of the equation is given by:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
Substituting the values of \( b \), \( D \), and \( a \):
\[ x = \frac{-(p^2 - q^2) \pm \sqrt{(p^2 + q^2)^2}}{2p^2} \]
Since \( \sqrt{(p^2 + q^2)^2} = p^2 + q^2 \), we have:
\[ x = \frac{-(p^2 - q^2) \pm (p^2 + q^2)}{2p^2} \]

### Step 4: Simplify the expressions
Calculating the two possible values for \( x \):
1. For the positive case:
\[ x_1 = \frac{-(p^2 - q^2) + (p^2 + q^2)}{2p^2} = \frac{2q^2}{2p^2} = \frac{q^2}{p^2} \]

2. For the negative case:
\[ x_2 = \frac{-(p^2 - q^2) - (p^2 + q^2)}{2p^2} = \frac{-2p^2}{2p^2} = -1 \]

### Final Result
The solutions to the equation \( p^2 x^2 + (p^2 - q^2)x - q^2 = 0 \) are:
\[ x = \frac{q^2}{p^2} \quad \text{and} \quad x = -1 \]

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