If `3x^(2) -17x+10 =0 and x^(2)-5x+m =0` has a common root, then sum of all possible real values of 'm' is:

Options

A. 0
B. `-26/9`
C. `29/9`
D. `26/3`

Answer

To solve the problem, we need to find the values of 'm' such that the quadratic equations \(3x^2 - 17x + 10 = 0\) and \(x^2 - 5x + m = 0\) have a common root.

### Step-by-Step Solution:

1. **Identify the first quadratic equation:**
\[ 3x^2 - 17x + 10 = 0 \]

2. **Factor the first quadratic equation:**
We need to find two numbers that multiply to \(3 \times 10 = 30\) and add to \(-17\). These numbers are \(-15\) and \(-2\).
\[ 3x^2 - 15x - 2x + 10 = 0 \]
Grouping the terms:
\[ 3x(x - 5) - 2(x - 5) = 0 \]
Factoring out \((x - 5)\):
\[ (x - 5)(3x - 2) = 0 \]

3. **Find the roots of the first equation:**
Setting each factor to zero gives:
\[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \]
\[ 3x - 2 = 0 \quad \Rightarrow \quad x = \frac{2}{3} \]
Thus, the roots are \(x = 5\) and \(x = \frac{2}{3}\).

4. **Consider the second quadratic equation:**
\[ x^2 - 5x + m = 0 \]
We need to find the values of \(m\) such that this equation has a common root with the first equation.

5. **Substituting the common root \(x = 5\):**
Substitute \(x = 5\) into the second equation:
\[ 5^2 - 5 \cdot 5 + m = 0 \]
Simplifying gives:
\[ 25 - 25 + m = 0 \quad \Rightarrow \quad m = 0 \]

6. **Substituting the common root \(x = \frac{2}{3}\):**
Substitute \(x = \frac{2}{3}\) into the second equation:
\[ \left(\frac{2}{3}\right)^2 - 5 \cdot \left(\frac{2}{3}\right) + m = 0 \]
This simplifies to:
\[ \frac{4}{9} - \frac{10}{3} + m = 0 \]
To combine the terms, convert \(-\frac{10}{3}\) to a fraction with a denominator of 9:
\[ -\frac{10}{3} = -\frac{30}{9} \]
Thus:
\[ \frac{4}{9} - \frac{30}{9} + m = 0 \quad \Rightarrow \quad -\frac{26}{9} + m = 0 \quad \Rightarrow \quad m = \frac{26}{9} \]

7. **Sum of all possible values of \(m\):**
The possible values of \(m\) are \(0\) and \(\frac{26}{9}\). Therefore, the sum is:
\[ 0 + \frac{26}{9} = \frac{26}{9} \]

### Final Answer:
The sum of all possible real values of \(m\) is:
\[ \frac{26}{9} \]