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

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

### Step 2: Write the quadratic formula
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

### Step 3: Substitute the coefficients into the formula
Substituting \( a \), \( b \), and \( c \) into the quadratic formula:
\[ x = \frac{-(p^2 - q^2) \pm \sqrt{(p^2 - q^2)^2 - 4(p^2)(-q^2)}}{2(p^2)} \]

### Step 4: Simplify the expression
First, simplify the numerator:
\[ -(p^2 - q^2) = -p^2 + q^2 = q^2 - p^2 \]
Now, simplify the discriminant:
\[ (p^2 - q^2)^2 - 4(p^2)(-q^2) = (p^2 - q^2)^2 + 4p^2q^2 \]
Expanding \( (p^2 - q^2)^2 \):
\[ = p^4 - 2p^2q^2 + q^4 + 4p^2q^2 = p^4 + 2p^2q^2 + q^4 = (p^2 + q^2)^2 \]
So, the expression becomes:
\[ x = \frac{q^2 - p^2 \pm \sqrt{(p^2 + q^2)^2}}{2p^2} \]

### Step 5: Simplify the square root
Since \( \sqrt{(p^2 + q^2)^2} = p^2 + q^2 \), we have:
\[ x = \frac{q^2 - p^2 \pm (p^2 + q^2)}{2p^2} \]

### Step 6: Split into two cases
Now we can split this into two cases:
1. \( x = \frac{(q^2 - p^2) + (p^2 + q^2)}{2p^2} = \frac{2q^2}{2p^2} = \frac{q^2}{p^2} \)
2. \( x = \frac{(q^2 - p^2) - (p^2 + q^2)}{2p^2} = \frac{-2p^2}{2p^2} = -1 \)

### Final Solutions
Thus, the solutions to the equation are:
\[ x = \frac{q^2}{p^2} \quad \text{and} \quad x = -1 \]

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