To determine which equation can be used to find the midpoint [tex]\( m \)[/tex] of two numbers whose product is -99 and are 10 units away from their midpoint, we need to follow certain logical steps.
### Step-by-Step Solution:
1. Express the Numbers in Terms of [tex]\( m \)[/tex]:
- Since the two numbers are 10 units away in different directions from their midpoint [tex]\( m \)[/tex], the numbers can be expressed as [tex]\( (m - 10) \)[/tex] and [tex]\( (m + 10) \)[/tex].
2. Set Up the Product Equation:
- According to the problem, the product of these two numbers is -99. Therefore, we write the following equation:
[tex]\[
(m - 10)(m + 10) = -99
\][/tex]
3. Simplify the Product Equation:
- We recognize that the left side of the equation is a difference of squares:
[tex]\[
(m - 10)(m + 10) = m^2 - 10^2
\][/tex]
- Simplify the expression:
[tex]\[
m^2 - 100 = -99
\][/tex]
4. Rewrite the Equation to Match Choices:
- The simplified form of the equation we obtained is:
[tex]\[
m^2 - 100 = -99
\][/tex]
- Notice that this matches the form of one of the given options exactly:
[tex]\[
(m-10)(m+10)=99
\][/tex]
- Inspect the other options to confirm that none of them represent this form:
- [tex]\((m-5)(m+5)=99\)[/tex] does not match our equation.
- [tex]\(m^2-25=-99\)[/tex] does not match our equation since [tex]\(25\)[/tex] is not [tex]\(100\)[/tex].
- [tex]\(m^2-100=-99\)[/tex] is our simplified equation but slightly rewritten.
5. Identify the Correct Answer:
- The equation [tex]\((m - 10)(m + 10) = 99\)[/tex] is directly derived from our analysis and matches the scenario of the problem perfectly.
Therefore, the correct equation that can be used to find [tex]\( m \)[/tex], the midpoint of the two numbers, is:
[tex]\[
\boxed{(m - 10)(m + 10) = 99}
\][/tex]
