What are the roots of the quadratic equation `21 x ^(2) - 37x - 28=0` ?

Options

A. `(-7)/(3), (4)/(7)`
B. `(3)/(7) , (-7)/(4)`
C. `(7)/(3) , (-4)/(7)`
D. `(-3)/(7) , (7)/(4)`

Answer

To find the roots of the quadratic equation \( 21x^2 - 37x - 28 = 0 \), we can use the factorization method. Here’s a step-by-step solution:

### Step 1: Identify the coefficients
The given quadratic equation is in the standard form \( ax^2 + bx + c = 0 \), where:
- \( a = 21 \)
- \( b = -37 \)
- \( c = -28 \)

### Step 2: Calculate the product \( ac \)
Calculate the product of \( a \) and \( c \):
\[ ac = 21 \times (-28) = -588 \]

### Step 3: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to \( -588 \) and add to \( -37 \). The numbers are \( -49 \) and \( 12 \) because:
\[ -49 \times 12 = -588 \quad \text{and} \quad -49 + 12 = -37 \]

### Step 4: Rewrite the equation
Rewrite the middle term using the two numbers found:
\[ 21x^2 - 49x + 12x - 28 = 0 \]

### Step 5: Factor by grouping
Group the terms:
\[ (21x^2 - 49x) + (12x - 28) = 0 \]
Factor out the common factors in each group:
\[ 7x(3x - 7) + 4(3x - 7) = 0 \]

### Step 6: Factor out the common binomial
Now, factor out the common binomial \( (3x - 7) \):
\[ (3x - 7)(7x + 4) = 0 \]

### Step 7: Set each factor to zero
Set each factor equal to zero:
1. \( 3x - 7 = 0 \)
2. \( 7x + 4 = 0 \)

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

For the second equation:
\[ 7x + 4 = 0 \implies 7x = -4 \implies x = -\frac{4}{7} \]

### Conclusion
The roots of the quadratic equation \( 21x^2 - 37x - 28 = 0 \) are:
\[ x = \frac{7}{3} \quad \text{and} \quad x = -\frac{4}{7} \]

---