Find the roots of the quadratic equation `27x ^(2) + 57 x - 14 =0`

Options

A. `-2//9, 7//3`
B. `2//9,-7//3`
C. `9//2, -3//7`
D. `-9//2,3//7`

Answer

To find the roots of the quadratic equation \( 27x^2 + 57x - 14 = 0 \), we can follow these steps:

### Step 1: Identify coefficients
The given quadratic equation is in the standard form \( ax^2 + bx + c = 0 \), where:
- \( a = 27 \)
- \( b = 57 \)
- \( c = -14 \)

### Step 2: Calculate the product \( ac \)
We need to find the product \( ac \):
\[ ac = 27 \times (-14) = -378 \]

### Step 3: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to \( -378 \) and add to \( 57 \). After testing possible pairs, we find:
- The numbers are \( 63 \) and \( -6 \) because:
- \( 63 \times (-6) = -378 \)
- \( 63 + (-6) = 57 \)

### Step 4: Rewrite the middle term
We can rewrite the equation using these two numbers:
\[ 27x^2 + 63x - 6x - 14 = 0 \]

### Step 5: Factor by grouping
Now, we group the terms:
\[ (27x^2 + 63x) + (-6x - 14) = 0 \]
Factoring each group gives:
\[ 9x(3x + 7) - 2(3x + 7) = 0 \]

### Step 6: Factor out the common binomial
Now we can factor out the common factor \( (3x + 7) \):
\[ (3x + 7)(9x - 2) = 0 \]

### Step 7: Set each factor to zero
Now we set each factor to zero:
1. \( 3x + 7 = 0 \)
2. \( 9x - 2 = 0 \)

### Step 8: Solve for \( x \)
For the first equation:
\[ 3x + 7 = 0 \implies 3x = -7 \implies x = -\frac{7}{3} \]

For the second equation:
\[ 9x - 2 = 0 \implies 9x = 2 \implies x = \frac{2}{9} \]

### Final Roots
The roots of the quadratic equation \( 27x^2 + 57x - 14 = 0 \) are:
\[ x = -\frac{7}{3} \quad \text{and} \quad x = \frac{2}{9} \]