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Difficulty: MediumPositive and Negative Number Properties

If mm and nn are real numbers such that m<0<nm < 0 < n and m2−3mn−4n2=0m^2 - 3mn - 4n^2 = 0, what is the value of m+2nm−n\frac{m + 2n}{m - n}?

Answer: -0.5

Answer

The value of the expression is -0.5.
Factoring m2−3mn−4n2=0m^2 - 3mn - 4n^2 = 0 yields (m−4n)(m+n)=0(m - 4n)(m + n) = 0, giving solutions m=4nm = 4n or m=−nm = -n. Given m<0<nm < 0 < n, mm is negative and nn is positive, so mm and nn must have opposite signs. The solution m=4nm = 4n implies that mm and nn have the same sign (both positive), which violates m<0m < 0. Therefore, mm must equal −n-n. Substituting m=−nm = -n into m+2nm−n\frac{m + 2n}{m - n} results in −n+2n−n−n=n−2n=−0.5\frac{-n + 2n}{-n - n} = \frac{n}{-2n} = -0.5.

Step-by-Step Solution

1
Factor the quadratic equation m2−3mn−4n2=0m^2 - 3mn - 4n^2 = 0.
(m−4n)(m+n)=0(m - 4n)(m + n) = 0
Factoring allows determination of the possible linear relationships between mm and nn.
2
Apply the sign condition m<0<nm < 0 < n to eliminate the invalid case.
m=−nm = -n
Since n>0n > 0, 4n4n must be positive. Setting m=4nm = 4n would make m>0m > 0, contradicting m<0m < 0. Thus, m+n=0m + n = 0, giving m=−nm = -n.
3
Substitute m=−nm = -n into m+2nm−n\frac{m + 2n}{m - n}.
\frac{-n + 2n}{-n - n} = \frac{n}{-2n} = -0.5
Replacing mm with −n-n simplifies the ratio to a constant value.

Key Concept

Positive and Negative Number Properties in Quadratic Constraints
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Topics

Positive and Negative Number Properties

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