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

Answer

To solve the quadratic equation \( a^2 x^2 + 2ax + 1 = 0 \), we will use the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

### Step 1: Identify coefficients
In the equation \( a^2 x^2 + 2ax + 1 = 0 \), we can identify the coefficients as follows:
- \( A = a^2 \) (coefficient of \( x^2 \))
- \( B = 2a \) (coefficient of \( x \))
- \( C = 1 \) (constant term)

### Step 2: Calculate the discriminant
The discriminant \( D \) is given by the formula:

\[ D = B^2 - 4AC \]

Substituting the values of \( A \), \( B \), and \( C \):

\[ D = (2a)^2 - 4(a^2)(1) \]
\[ D = 4a^2 - 4a^2 \]
\[ D = 0 \]

### Step 3: Apply the quadratic formula
Since the discriminant \( D = 0 \), we will have one real and repeated root. Now, we can substitute the values into the quadratic formula:

\[ x = \frac{-B \pm \sqrt{D}}{2A} \]

Substituting \( B = 2a \), \( D = 0 \), and \( A = a^2 \):

\[ x = \frac{-2a \pm \sqrt{0}}{2a^2} \]
\[ x = \frac{-2a}{2a^2} \]

### Step 4: Simplify the expression
Now, simplify the expression:

\[ x = \frac{-2a}{2a^2} = \frac{-1}{a} \]

### Conclusion
Thus, the solution to the equation \( a^2 x^2 + 2ax + 1 = 0 \) is:

\[ x = -1 \]