Solve the following quadratic equations by factorisation method :
`x^(2)-5x-36=0`

Answer

To solve the quadratic equation \( x^2 - 5x - 36 = 0 \) by the factorization method, follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation is in the standard form \( ax^2 + bx + c = 0 \), where:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = -5 \) (coefficient of \( x \))
- \( c = -36 \) (constant term)

### Step 2: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers \( d_1 \) and \( d_2 \) such that:
- \( d_1 \times d_2 = ac = 1 \times (-36) = -36 \)
- \( d_1 + d_2 = b = -5 \)

After checking the pairs of factors of \(-36\):
- The pair \(-9\) and \(4\) works because:
- \(-9 \times 4 = -36\)
- \(-9 + 4 = -5\)

### Step 3: Rewrite the middle term using the two numbers
We can rewrite the equation by splitting the middle term:
\[ x^2 - 9x + 4x - 36 = 0 \]

### Step 4: Factor by grouping
Now, group the terms:
\[ (x^2 - 9x) + (4x - 36) = 0 \]
Factor out the common terms in each group:
\[ x(x - 9) + 4(x - 9) = 0 \]

### Step 5: Factor out the common binomial
Now, we can factor out the common binomial \((x - 9)\):
\[ (x - 9)(x + 4) = 0 \]

### Step 6: Set each factor to zero
Now, set each factor equal to zero to find the roots:
1. \( x - 9 = 0 \) → \( x = 9 \)
2. \( x + 4 = 0 \) → \( x = -4 \)

### Step 7: Write the final solution
The solutions to the quadratic equation \( x^2 - 5x - 36 = 0 \) are:
\[ x = 9 \quad \text{and} \quad x = -4 \]

---