Question

two numbers are 10 units away in different directions from their midpoint, m, on a number line. the product of the numbers is -99. which equation can be used to find m, the midpoint of the two numbers? (m - 5)(m + 5) = 99 (m - 10)(m + 10) = 99 m² - 25 = -99 m² - 100 = -99

Explanation:

Step1: Define the two numbers

Since the two numbers are 10 units away in different directions from their midpoint \( m \), one number is \( m + 5 \) and the other is \( m - 5 \) (because 10 units split into two equal parts is 5 units each).

Step2: Set up the product equation

The product of the two numbers is -99, so we have \((m - 5)(m + 5)=- 99\). Using the difference of squares formula \( (a - b)(a + b)=a^{2}-b^{2} \), this simplifies to \( m^{2}-25=-99 \).

Step3: Analyze the options

• Option 1: \((m - 5)(m + 5) = 99\) is incorrect because the product is -99, not 99.
• Option 2: \((m - 10)(m + 10)=99\) is incorrect because the distance from the midpoint is 5 units (since 10 units total in both directions), not 10 units.
• Option 3: \(m^{2}-25=-99\) matches our derived equation.
• Option 4: \(m^{2}-100=-99\) is incorrect as the difference of squares should involve 25 (from \(5^{2}\)), not 100 (from \(10^{2}\)).

Answer:

\( m^{2}-25 = -99 \) (the third option)