The range of values of m for which the equation `(m-5) x^(2)+2(m-10) x+m+10=0` has real roots of the same sign, is given by

Options

A. `m gt 10`
B. `-5 lt m lt 5`
C. `m lt -10,5 lt m le 6`
D. none of these

Answer

To find the range of values of \( m \) for which the equation

\[ (m-5)x^2 + 2(m-10)x + (m+10) = 0 \]

has real roots of the same sign, we will follow these steps:

### Step 1: Identify the coefficients
From the quadratic equation \( ax^2 + bx + c = 0 \), we have:
- \( a = m - 5 \)
- \( b = 2(m - 10) \)
- \( c = m + 10 \)

### Step 2: Condition for real roots
For the roots to be real, the discriminant must be non-negative:

\[ D = b^2 - 4ac \geq 0 \]

Substituting the values of \( a \), \( b \), and \( c \):

\[ D = [2(m - 10)]^2 - 4(m - 5)(m + 10) \geq 0 \]

### Step 3: Simplify the discriminant
Calculating \( D \):

\[ D = 4(m - 10)^2 - 4(m - 5)(m + 10) \]

Expanding both terms:

1. \( 4(m - 10)^2 = 4(m^2 - 20m + 100) = 4m^2 - 80m + 400 \)
2. \( 4(m - 5)(m + 10) = 4(m^2 + 10m - 5m - 50) = 4(m^2 + 5m - 50) = 4m^2 + 20m - 200 \)

Now, substituting back into the discriminant:

\[ D = (4m^2 - 80m + 400) - (4m^2 + 20m - 200) \]

This simplifies to:

\[ D = -80m + 400 - 20m + 200 = -100m + 600 \]

Setting the discriminant greater than or equal to zero:

\[ -100m + 600 \geq 0 \]

### Step 4: Solve the inequality
Rearranging gives:

\[ 100m \leq 600 \implies m \leq 6 \]

### Step 5: Condition for roots of the same sign
For the roots to have the same sign, the product of the roots (given by \( \frac{c}{a} \)) must be positive:

\[ \frac{m + 10}{m - 5} > 0 \]

### Step 6: Analyze the sign of the product
This inequality holds when both the numerator and denominator are either both positive or both negative.

1. **Case 1:** Both positive
- \( m + 10 > 0 \implies m > -10 \)
- \( m - 5 > 0 \implies m > 5 \)

Thus, from this case, we have \( m > 5 \).

2. **Case 2:** Both negative
- \( m + 10 < 0 \implies m < -10 \)
- \( m - 5 < 0 \implies m < 5 \)

Thus, from this case, we have \( m < -10 \).

### Step 7: Combine the conditions
From the discriminant condition, we have \( m \leq 6 \). Combining this with the conditions for the product of the roots, we get:

- \( m < -10 \)
- \( 5 < m \leq 6 \)

### Final Answer
The range of values of \( m \) for which the equation has real roots of the same sign is:

\[ m \in (-\infty, -10) \cup (5, 6] \]