Problem 14 · 2025 Math Kangaroo
Hard
Algebra & Patterns
evaluate-formulacasework
A student draws the graphs of two linear functions in a coordinate system as shown. What is certain about the expression \(ab + cd - (ac + bd)\)?
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Answer: C — It is positive.
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Hint 1 of 2
Factor the expression ab + cd − ac − bd.
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Hint 2 of 2
It equals (a − d)(b − c); read the signs of a, b, c, d off the graph.
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Approach: factor, then read signs from the graph
- ab + cd − ac − bd = (a − d)(b − c).
- From the graph the rising line gives a > 0, b > 0; the falling line gives c < 0, d < 0.
- Then b − c > 0 and a − d > 0, so the product is positive.
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