Grades 11-12 Video Solutions 2023-2023_11-12

Video Transcription

13. The graphs of the function y equals x cubed plus 3x squared plus ax plus 2a plus 4 all pass through the same point, no matter what value of a is chosen. What is the sum  of the coordinates of that point? Let the common point be m comma n. Then we're just trying to find the answer m plus n. So  when we plug in x is equal to m into our function, y must always equal n. We shouldn't get any  other value for y. And so that basically means that whatever value of a we have, that shouldn't  matter. If x equals m, y is going to equal n. Well, that happens when x is equal to negative 2. As if we have that, then a times negative 2 in the original equation, when x equals  negative 2 plus 2a is equal to zero and a does not matter. Well, if a does not matter, then we have what we're looking for and x should be equal to negative 2. And when x  is equal to negative 2, we can solve for the value of y. And what we get is that y is equal to, after simplifying the algebra, negative 8 plus 12 minus 2a plus 2a, so you can see  the a's cancel out, plus 4. So y is equal to 8. So our final answer, m comma n, is equal to negative 2 comma 8. And our answer is the sum of those two values. So negative 2 plus  8 gives us our answer of 6. But wait a second, 6 is not any one of the answer choices. But we still have answer choice e, which says none of the above, and so we select that and  move on.

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