Solve each of the following equations :
`a^(2)x^(2)+2ax+1=0`

Answer

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

### Step 1: Identify the coefficients
The given equation is in the standard form of a quadratic equation \( ax^2 + bx + c = 0 \). Here, we identify:
- \( A = a^2 \)
- \( B = 2a \)
- \( C = 1 \)

### Step 2: Factor the quadratic equation
We can attempt to factor the quadratic equation. We want to express \( 2ax \) as a sum of two terms that multiply to \( a^2 \cdot 1 \) (which is \( a^2 \)).

We can rewrite \( 2ax \) as \( ax + ax \):
\[ a^2x^2 + ax + ax + 1 = 0 \]

### Step 3: Group the terms
Now, we can group the terms:
\[ (a^2x^2 + ax) + (ax + 1) = 0 \]

### Step 4: Factor by grouping
Next, we factor out the common terms from each group:
\[ ax(a x + 1) + 1(a x + 1) = 0 \]

Now we can factor out \( (ax + 1) \):
\[ (ax + 1)(ax + 1) = 0 \]

### Step 5: Set each factor to zero
Now we set each factor equal to zero:
\[ ax + 1 = 0 \]

### Step 6: Solve for \( x \)
Solving for \( x \):
\[ ax = -1 \implies x = -\frac{1}{a} \]

### Conclusion
The solution to the quadratic equation \( a^2x^2 + 2ax + 1 = 0 \) is:
\[ x = -\frac{1}{a} \]