Problem 19 · 2023 Math Kangaroo
Stretch
Algebra & Patterns
substitution
The graphs of the functions \(y = x^3 + 3x^2 + ax + 2a + 4\) all pass through a common point independent of the choice of a. How big is the sum of the co-ordinates of this common point?
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Answer: E — another number
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Hint 1 of 2
Group the terms that contain a together.
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Hint 2 of 2
The point must work for every a, so the coefficient of a has to vanish.
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Approach: make the coefficient of the parameter zero
- Write y = (x3+3x2+4) + a(x+2); for this to be independent of a, set x+2 = 0, so x = −2.
- Then y = (−8 + 12 + 4) = 8, giving the fixed point (−2, 8).
- Sum of coordinates = −2 + 8 = 6, which is not among the first four options, so the answer is another number.
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