The two roots of a quadratic equation are given as `x =5/3 and x=(-3)/10.` The equation can be writing as:
A)`(10x – 3)(3x - 5) = 0`
B)`(10x+3)(3x - 5) = 0`
C)`(10x +3)(3x + 5) = 0`
D)`(10x - 3)(3x + 5) = 0`

Options

A. `(10x – 3)(3x - 5) = 0`
B. `(10x+3)(3x - 5) = 0`
C. `(10x +3)(3x + 5) = 0`
D. `(10x - 3)(3x + 5) = 0`

Answer

To find the quadratic equation from the given roots \( x = \frac{5}{3} \) and \( x = -\frac{3}{10} \), we can follow these steps:

### Step 1: Write the factors from the roots
The roots of a quadratic equation can be expressed in factorized 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, we have:
\[ x - \frac{5}{3} = 0 \quad \text{and} \quad x + \frac{3}{10} = 0 \]

### Step 2: Rewrite the factors
We can rewrite the factors as:
\[ \left(x - \frac{5}{3}\right) \quad \text{and} \quad \left(x + \frac{3}{10}\right) \]

### Step 3: Eliminate the fractions
To eliminate the fractions, we can multiply each factor by the least common multiple (LCM) of the denominators. The denominators are 3 and 10, and their LCM is 30. Thus, we multiply the first factor by 30:
\[ 30\left(x - \frac{5}{3}\right) = 30x - 50 \]
And the second factor by 30:
\[ 30\left(x + \frac{3}{10}\right) = 30x + 9 \]

### Step 4: Write the full equation
Now, we can write the equation as:
\[ (30x - 50)(30x + 9) = 0 \]

### Step 5: Simplify the equation
Now we can simplify this equation. However, we want to express it in a form that matches the options given. We can factor out common multiples:
\[ (10x - 5)(3x + 3) = 0 \]
This can be rearranged to:
\[ (10x - 3)(3x - 5) = 0 \]

### Step 6: Identify the correct option
Now we can compare this with the options provided:
- A) \((10x - 3)(3x - 5) = 0\)
- B) \((10x + 3)(3x - 5) = 0\)
- C) \((10x + 3)(3x + 5) = 0\)
- D) \((10x - 3)(3x + 5) = 0\)

The correct option is:
**A) \((10x - 3)(3x - 5) = 0\)**