Solve each of the following quadratic equations:
`sqrt(2)x^(2)+7x+5sqrt(2)=0`

Explanation

To solve the quadratic equation \( \sqrt{2}x^2 + 7x + 5\sqrt{2} = 0 \), we will follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation is in the standard form \( ax^2 + bx + c = 0 \).
Here,
- \( a = \sqrt{2} \)
- \( b = 7 \)
- \( c = 5\sqrt{2} \)

### Step 2: Calculate the product \( ac \)
We need to find the product of \( a \) and \( c \):
\[ ac = \sqrt{2} \times 5\sqrt{2} = 5 \times 2 = 10 \]

### Step 3: Find two numbers that multiply to \( ac \) and add to \( b \)
We need two numbers that multiply to \( 10 \) (the value of \( ac \)) and add up to \( 7 \) (the value of \( b \)). The numbers that satisfy these conditions are \( 5 \) and \( 2 \) because:
\[ 5 \times 2 = 10 \quad \text{and} \quad 5 + 2 = 7 \]

### Step 4: Rewrite the middle term
Now, we can rewrite the equation by splitting the middle term using the numbers we found:
\[ \sqrt{2}x^2 + 5x + 2x + 5\sqrt{2} = 0 \]

### Step 5: Factor by grouping
Next, we group the terms:
\[ (\sqrt{2}x^2 + 5x) + (2x + 5\sqrt{2}) = 0 \]
Factoring out common factors from each group:
\[ x(\sqrt{2}x + 5) + \sqrt{2}(2x + 5) = 0 \]
Now we can factor out \( (\sqrt{2}x + 5) \):
\[ (\sqrt{2}x + 5)(x + \sqrt{2}) = 0 \]

### Step 6: Set each factor to zero
Now, we set each factor equal to zero:
1. \( \sqrt{2}x + 5 = 0 \)
2. \( x + \sqrt{2} = 0 \)

### Step 7: Solve for \( x \)
For the first factor:
\[ \sqrt{2}x + 5 = 0 \implies \sqrt{2}x = -5 \implies x = -\frac{5}{\sqrt{2}} = -\frac{5\sqrt{2}}{2} \]

For the second factor:
\[ x + \sqrt{2} = 0 \implies x = -\sqrt{2} \]

### Final Solution
The roots of the quadratic equation are:
\[ x = -\frac{5\sqrt{2}}{2} \quad \text{and} \quad x = -\sqrt{2} \]