The roots of the quadratic equation `3x^2 - 5x + 2 = 0` are

Options

A. `(2/3,1)`
B. `(1,1)`
C. `(1/3, 1)`
D. none of these

Answer

To find the roots of the quadratic equation \(3x^2 - 5x + 2 = 0\), we can use the quadratic formula or factorization. Here, we will use the factorization method.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given quadratic equation is in the form \(ax^2 + bx + c = 0\), where:
- \(a = 3\)
- \(b = -5\)
- \(c = 2\)

2. **Calculate the product \(ac\)**:
We need to find two numbers that multiply to \(ac\) (which is \(3 \times 2 = 6\)) and add up to \(b\) (which is \(-5\)).
\[ ac = 3 \times 2 = 6 \]

3. **Find two numbers**:
We are looking for two numbers that add to \(-5\) and multiply to \(6\). The numbers are \(-2\) and \(-3\) because:
\[ -2 + (-3) = -5 \quad \text{and} \quad -2 \times -3 = 6 \]

4. **Rewrite the middle term**:
We can rewrite the equation by splitting the middle term using the two numbers found:
\[ 3x^2 - 2x - 3x + 2 = 0 \]

5. **Group the terms**:
Now, we group the terms:
\[ (3x^2 - 2x) + (-3x + 2) = 0 \]

6. **Factor by grouping**:
Factor out the common terms from each group:
\[ x(3x - 2) - 1(3x - 2) = 0 \]
This can be factored as:
\[ (3x - 2)(x - 1) = 0 \]

7. **Set each factor to zero**:
Now, we set each factor equal to zero:
\[ 3x - 2 = 0 \quad \text{or} \quad x - 1 = 0 \]

8. **Solve for \(x\)**:
For \(3x - 2 = 0\):
\[ 3x = 2 \quad \Rightarrow \quad x = \frac{2}{3} \]
For \(x - 1 = 0\):
\[ x = 1 \]

9. **Conclusion**:
The roots of the quadratic equation \(3x^2 - 5x + 2 = 0\) are:
\[ x = \frac{2}{3} \quad \text{and} \quad x = 1 \]

### Final Answer:
The roots are \(x = \frac{2}{3}\) and \(x = 1\).

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