If the roots of `x^(2)-2(7+3m)x+55m+45=0`, are equal, then m =

Options

A. 1, `(4)/(9)`
B. -1, `(-4)/(9)`
C. 1, `(-4)/(9)`
D. -1, `(+4)/(9)`

Answer

To solve the problem, we need to find the value of \( m \) such that the roots of the quadratic equation \( x^2 - 2(7 + 3m)x + (55m + 45) = 0 \) are equal. For the roots of a quadratic equation to be equal, the discriminant must be zero.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given quadratic equation is:
\[ x^2 - 2(7 + 3m)x + (55m + 45) = 0 \]
Here, \( a = 1 \), \( b = -2(7 + 3m) \), and \( c = 55m + 45 \).

2. **Set up the discriminant**:
The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[ D = b^2 - 4ac \]
For the roots to be equal, we set \( D = 0 \):
\[ (-2(7 + 3m))^2 - 4(1)(55m + 45) = 0 \]

3. **Calculate \( b^2 \)**:
\[ b^2 = (-2(7 + 3m))^2 = 4(7 + 3m)^2 \]

4. **Calculate \( 4ac \)**:
\[ 4ac = 4 \cdot 1 \cdot (55m + 45) = 4(55m + 45) = 220m + 180 \]

5. **Set up the equation**:
Now substituting \( b^2 \) and \( 4ac \) into the discriminant equation:
\[ 4(7 + 3m)^2 - (220m + 180) = 0 \]

6. **Expand \( (7 + 3m)^2 \)**:
\[ (7 + 3m)^2 = 49 + 42m + 9m^2 \]
Therefore,
\[ 4(49 + 42m + 9m^2) = 196 + 168m + 36m^2 \]

7. **Combine the equation**:
Substitute back into the equation:
\[ 196 + 168m + 36m^2 - (220m + 180) = 0 \]
Simplifying this gives:
\[ 36m^2 + 168m - 220m + 196 - 180 = 0 \]
\[ 36m^2 - 52m + 16 = 0 \]

8. **Divide by 4**:
To simplify, divide the entire equation by 4:
\[ 9m^2 - 13m + 4 = 0 \]

9. **Use the quadratic formula**:
The quadratic formula is:
\[ m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here \( a = 9 \), \( b = -13 \), and \( c = 4 \):
\[ m = \frac{13 \pm \sqrt{(-13)^2 - 4 \cdot 9 \cdot 4}}{2 \cdot 9} \]
\[ = \frac{13 \pm \sqrt{169 - 144}}{18} \]
\[ = \frac{13 \pm \sqrt{25}}{18} \]
\[ = \frac{13 \pm 5}{18} \]

10. **Calculate the two possible values of \( m \)**:
- First value:
\[ m = \frac{13 + 5}{18} = \frac{18}{18} = 1 \]
- Second value:
\[ m = \frac{13 - 5}{18} = \frac{8}{18} = \frac{4}{9} \]

### Final Answer:
Thus, the values of \( m \) for which the roots are equal are:
\[ m = 1 \quad \text{or} \quad m = \frac{4}{9} \]