Solve for x by fractorisation
`3sqrt5x ^(2) + 25 x + 10 sqrt5=0`

Answer

To solve the quadratic equation \(3\sqrt{5}x^2 + 25x + 10\sqrt{5} = 0\) by factorization, 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:
- \(a = 3\sqrt{5}\)
- \(b = 25\)
- \(c = 10\sqrt{5}\)

### Step 2: Multiply \(a\) and \(c\)
We need to find the product of \(a\) and \(c\):
\[ ac = (3\sqrt{5})(10\sqrt{5}) = 30 \cdot 5 = 150 \]

### Step 3: Find two numbers that multiply to \(ac\) and add to \(b\)
We need two numbers that multiply to \(150\) and add up to \(25\). The numbers \(30\) and \(-5\) satisfy this condition:
\[ 30 \times (-5) = -150 \quad \text{(not applicable)} \]
Instead, we can use \(30\) and \(-5\):
\[ 30 + (-5) = 25 \quad \text{(correct)} \]

### Step 4: Rewrite the middle term
We can rewrite the equation by splitting the middle term using the two numbers found:
\[ 3\sqrt{5}x^2 + 30x - 5x + 10\sqrt{5} = 0 \]

### Step 5: Factor by grouping
Now, we group the terms:
\[ (3\sqrt{5}x^2 + 30x) + (-5x + 10\sqrt{5}) = 0 \]

Now, factor out the common terms from each group:
1. From the first group, factor out \(3\sqrt{5}x\):
\[ 3\sqrt{5}x(x + 10) \]
2. From the second group, factor out \(-5\):
\[ -5(x - 2\sqrt{5}) \]

So we can rewrite the equation as:
\[ 3\sqrt{5}x(x + 10) - 5(x - 2\sqrt{5}) = 0 \]

### Step 6: Combine the factors
Now we can combine the factors:
\[ (3\sqrt{5}x - 5)(x + 2\sqrt{5}) = 0 \]

### Step 7: Set each factor to zero
Now we can set each factor to zero:
1. \(3\sqrt{5}x - 5 = 0\)
2. \(x + 2\sqrt{5} = 0\)

### Step 8: Solve for \(x\)
1. From \(3\sqrt{5}x - 5 = 0\):
\[ 3\sqrt{5}x = 5 \implies x = \frac{5}{3\sqrt{5}} = \frac{\sqrt{5}}{3} \]

2. From \(x + 2\sqrt{5} = 0\):
\[ x = -2\sqrt{5} \]

### Final Solutions
The solutions for \(x\) are:
\[ x = \frac{\sqrt{5}}{3} \quad \text{and} \quad x = -2\sqrt{5} \]

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