Find the roots of following quadratic equation by factorisation: `sqrt(2)x^(2)+7x+5sqrt(2)=0`

Answer

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

### Step 1: Rewrite the equation
We start with the equation:
\[ \sqrt{2}x^2 + 7x + 5\sqrt{2} = 0 \]

### Step 2: Factor the quadratic expression
To factor the quadratic, we need to express \( 7x \) as a sum of two terms whose coefficients multiply to \( \sqrt{2} \times 5\sqrt{2} = 10 \) (the product of the coefficient of \( x^2 \) and the constant term) and add up to \( 7 \) (the coefficient of \( x \)).

We can break \( 7x \) into \( 2x + 5x \):
\[ \sqrt{2}x^2 + 2x + 5x + 5\sqrt{2} = 0 \]

### Step 3: Group the terms
Now, we can group the terms:
\[ (\sqrt{2}x^2 + 2x) + (5x + 5\sqrt{2}) = 0 \]

### Step 4: Factor by grouping
Factor out common terms from each group:
\[ x(\sqrt{2}x + 2) + 5(\sqrt{2}x + 2) = 0 \]

Now we can factor out \( \sqrt{2}x + 2 \):
\[ (\sqrt{2}x + 2)(x + 5) = 0 \]

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

### Step 6: Solve for \( x \)
From the first equation:
\[ \sqrt{2}x + 2 = 0 \implies \sqrt{2}x = -2 \implies x = -\frac{2}{\sqrt{2}} = -\sqrt{2} \]

From the second equation:
\[ x + 5 = 0 \implies x = -5 \]

### Step 7: State the roots
The roots of the quadratic equation are:
\[ x = -\sqrt{2}, \quad x = -5 \]

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