If `x^(2) - (5m - 2)x + 4m^(2) + 10m + 25` is a perfect square then m =

Options

A. `8/3` or `4`
B. `-8/3` or `4`
C. `4/3` or `8`
D. `-4/3` or `8`

Answer

To determine the value of \( m \) such that the quadratic expression \( x^2 - (5m - 2)x + (4m^2 + 10m + 25) \) is a perfect square, we need to ensure that the discriminant of the quadratic equation is zero.

### Step-by-step Solution:

1. **Identify the coefficients**:
The quadratic expression can be written in the standard form \( ax^2 + bx + c \):
- \( a = 1 \)
- \( b = -(5m - 2) = -5m + 2 \)
- \( c = 4m^2 + 10m + 25 \)

2. **Calculate the discriminant**:
The discriminant \( D \) is given by the formula:
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = (-5m + 2)^2 - 4(1)(4m^2 + 10m + 25) \]

3. **Expand the discriminant**:
First, calculate \( (-5m + 2)^2 \):
\[ (-5m + 2)^2 = 25m^2 - 20m + 4 \]
Now calculate \( 4(4m^2 + 10m + 25) \):
\[ 4(4m^2 + 10m + 25) = 16m^2 + 40m + 100 \]
Now substitute back into the discriminant:
\[ D = (25m^2 - 20m + 4) - (16m^2 + 40m + 100) \]

4. **Simplify the discriminant**:
Combine like terms:
\[ D = 25m^2 - 16m^2 - 20m - 40m + 4 - 100 \]
\[ D = 9m^2 - 60m - 96 \]

5. **Set the discriminant to zero**:
For the quadratic to be a perfect square, we set \( D = 0 \):
\[ 9m^2 - 60m - 96 = 0 \]

6. **Divide the equation by 3**:
Simplifying gives:
\[ 3m^2 - 20m - 32 = 0 \]

7. **Use the quadratic formula**:
The quadratic formula is given by:
\[ m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 3 \), \( b = -20 \), and \( c = -32 \):
\[ m = \frac{20 \pm \sqrt{(-20)^2 - 4 \cdot 3 \cdot (-32)}}{2 \cdot 3} \]

8. **Calculate the discriminant inside the square root**:
\[ (-20)^2 = 400 \]
\[ -4 \cdot 3 \cdot (-32) = 384 \]
Thus:
\[ m = \frac{20 \pm \sqrt{400 + 384}}{6} \]
\[ m = \frac{20 \pm \sqrt{784}}{6} \]
\[ m = \frac{20 \pm 28}{6} \]

9. **Calculate the two possible values for \( m \)**:
- First value:
\[ m = \frac{48}{6} = 8 \]
- Second value:
\[ m = \frac{-8}{6} = -\frac{4}{3} \]

10. **Final values of \( m \)**:
The values of \( m \) that make the quadratic a perfect square are:
\[ m = 8 \quad \text{and} \quad m = -\frac{4}{3} \]