Find the roots of the quadratic equation `sqrt2 x^(2) + 7x + 5 sqrt2 = 0`.

Answer

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

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

### Step 2: Use the quadratic formula
The roots of a quadratic equation can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

### Step 3: Calculate the discriminant
First, we need to calculate the discriminant \( D = b^2 - 4ac \):
\[ D = 7^2 - 4 \cdot \sqrt{2} \cdot 5\sqrt{2} \]
\[ D = 49 - 4 \cdot 2 \cdot 5 \]
\[ D = 49 - 40 \]
\[ D = 9 \]

### Step 4: Substitute values into the quadratic formula
Now we can substitute \( a \), \( b \), and \( D \) into the quadratic formula:
\[ x = \frac{-7 \pm \sqrt{9}}{2\sqrt{2}} \]
\[ x = \frac{-7 \pm 3}{2\sqrt{2}} \]

### Step 5: Calculate the two possible values for \( x \)
1. For the positive root:
\[ x_1 = \frac{-7 + 3}{2\sqrt{2}} = \frac{-4}{2\sqrt{2}} = \frac{-2}{\sqrt{2}} = -\sqrt{2} \]

2. For the negative root:
\[ x_2 = \frac{-7 - 3}{2\sqrt{2}} = \frac{-10}{2\sqrt{2}} = \frac{-5}{\sqrt{2}} = -\frac{5\sqrt{2}}{2} \]

### Step 6: Final roots
Thus, the roots of the quadratic equation \( \sqrt{2} x^2 + 7x + 5\sqrt{2} = 0 \) are:
\[ x_1 = -\sqrt{2}, \quad x_2 = -\frac{5\sqrt{2}}{2} \]

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