Solve each of the following equatins :
`a^(2)x^(2)+(a^(2)+b^(2))x+b^(2)=0`

Answer

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

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

### Step 2: Factor the quadratic equation
We can rewrite the equation by grouping terms:
\[ a^2 x^2 + (a^2 + b^2)x + b^2 = 0 \]
This can be rearranged as:
\[ a^2 x^2 + a^2 x + b^2 x + b^2 = 0 \]
Now, we can factor by grouping:
\[ a^2 x(x + 1) + b^2(x + 1) = 0 \]
This gives us:
\[ (a^2 x + b^2)(x + 1) = 0 \]

### Step 3: Set each factor to zero
Now, we set each factor equal to zero:
1. \( a^2 x + b^2 = 0 \)
2. \( x + 1 = 0 \)

### Step 4: Solve for \( x \)
From the first factor:
\[ a^2 x + b^2 = 0 \implies a^2 x = -b^2 \implies x = -\frac{b^2}{a^2} \]

From the second factor:
\[ x + 1 = 0 \implies x = -1 \]

### Step 5: Write the final solution
The solutions to the equation are:
\[ x = -\frac{b^2}{a^2}, \quad x = -1 \]

### Final Answer
Thus, the final solutions are:
\[ x = -1, \quad x = -\frac{b^2}{a^2} \]

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