Solve for x
`x ^(2) + 5 sqrt5x - 70 =0`

Answer

To solve the quadratic equation \( x^2 + 5\sqrt{5}x - 70 = 0 \) using the factorization method, we will follow these steps:

### Step 1: Identify the coefficients
The given equation is in the standard form of a quadratic equation \( ax^2 + bx + c = 0 \).
Here, \( a = 1 \), \( b = 5\sqrt{5} \), and \( c = -70 \).

### Step 2: Multiply \( a \) and \( c \)
We need to multiply \( a \) and \( c \):
\[ ac = 1 \times (-70) = -70 \]

### Step 3: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to \(-70\) and add to \( 5\sqrt{5} \).

Let's consider the factors of \(-70\):
- The pairs of factors are: \( (1, -70), (-1, 70), (2, -35), (-2, 35), (5, -14), (-5, 14), (7, -10), (-7, 10) \).

We need to find a combination that adds up to \( 5\sqrt{5} \). After testing various pairs, we find:
\[ 7\sqrt{5} \text{ and } -2\sqrt{5} \]
These two numbers multiply to \(-70\) and add to \( 5\sqrt{5} \).

### Step 4: Rewrite the middle term
We can rewrite the equation as:
\[ x^2 + 7\sqrt{5}x - 2\sqrt{5}x - 70 = 0 \]

### Step 5: Factor by grouping
Now, we group the terms:
\[ (x^2 + 7\sqrt{5}x) + (-2\sqrt{5}x - 70) = 0 \]
Factoring out the common terms:
\[ x(x + 7\sqrt{5}) - 2\sqrt{5}(x + 7\sqrt{5}) = 0 \]

### Step 6: Factor out the common binomial
Now we factor out the common binomial \( (x + 7\sqrt{5}) \):
\[ (x + 7\sqrt{5})(x - 2\sqrt{5}) = 0 \]

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

### Step 8: Final solutions
Thus, the solutions for the equation \( x^2 + 5\sqrt{5}x - 70 = 0 \) are:
\[ x = -7\sqrt{5} \quad \text{and} \quad x = 2\sqrt{5} \]

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