The two roots of a quadratic equation are given as `x = 4/3` and `x = -3/7`. The equation can be written as:

Options

A. `(7x-3) (3x-4) =0`
B. `(7x+3) (3x-4)=0`
C. `(7x + 3)(3x + 4)=0`
D. `(7x-3) (3x+4)=0`

Answer

To find the quadratic equation given the roots \( x = \frac{4}{3} \) and \( x = -\frac{3}{7} \), we can follow these steps:

### Step 1: Write the factors based on the roots
The roots of a quadratic equation can be expressed in factor form. If \( r_1 \) and \( r_2 \) are the roots, the factors can be written as:
\[ (x - r_1)(x - r_2) = 0 \]
For our roots:
- \( r_1 = \frac{4}{3} \)
- \( r_2 = -\frac{3}{7} \)

Thus, the factors become:
\[ \left(x - \frac{4}{3}\right) \text{ and } \left(x + \frac{3}{7}\right) \]

### Step 2: Convert the factors to eliminate fractions
To eliminate the fractions, we can multiply each factor by the denominators:
1. For \( x - \frac{4}{3} \), multiply by 3:
\[ 3\left(x - \frac{4}{3}\right) = 3x - 4 \]
2. For \( x + \frac{3}{7} \), multiply by 7:
\[ 7\left(x + \frac{3}{7}\right) = 7x + 3 \]

### Step 3: Write the equation
Now we can write the quadratic equation by multiplying the two factors:
\[ (3x - 4)(7x + 3) = 0 \]

### Step 4: Expand the equation
Now, we expand the product:
\[ 3x \cdot 7x + 3 \cdot 3x - 4 \cdot 7x - 4 \cdot 3 \]
This simplifies to:
\[ 21x^2 + 9x - 28x - 12 = 0 \]
Combining like terms gives:
\[ 21x^2 - 19x - 12 = 0 \]

### Final Result
Thus, the quadratic equation is:
\[ 21x^2 - 19x - 12 = 0 \]

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