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

Explanation

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

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

### Step 2: Calculate the product and sum
We need to find two numbers that multiply to \(A \cdot C = 2 \cdot (-a^2) = -2a^2\) and add up to \(B = a\).

### Step 3: Find the factors
We are looking for two numbers \(m\) and \(n\) such that:
- \(m \cdot n = -2a^2\)
- \(m + n = a\)

After testing possible pairs, we find that:
- \(m = 2a\)
- \(n = -a\)

### Step 4: Rewrite the middle term
We can rewrite the equation using these factors:
\[ 2x^2 + 2ax - ax - a^2 = 0 \]

### Step 5: Group the terms
Now, we can group the terms:
\[ (2x^2 + 2ax) + (-ax - a^2) = 0 \]

### Step 6: Factor by grouping
Factoring out the common terms:
\[ 2x(x + a) - a(x + a) = 0 \]

### Step 7: Factor out the common binomial
We can factor out \((x + a)\):
\[ (2x - a)(x + a) = 0 \]

### Step 8: Set each factor to zero
Now we set each factor equal to zero:
1. \(2x - a = 0\)
2. \(x + a = 0\)

### Step 9: Solve for \(x\)
From the first equation:
\[ 2x = a \implies x = \frac{a}{2} \]
From the second equation:
\[ x = -a \]

### Final Solution
The roots of the quadratic equation \(2x^2 + ax - a^2 = 0\) are:
\[ x = -a \quad \text{and} \quad x = \frac{a}{2} \]

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