Difference between revisions of "2024 AMC 12B Problems/Problem 13"
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Adding up the first and second statement, we get h+k with: | Adding up the first and second statement, we get h+k with: | ||
| − | =2x^2 + 2y^2 - 16x - 4y | + | |
| + | = 2x^2 + 2y^2 - 16x - 4y | ||
= 2(x^2 - 8x) + 2(y^2 - 2y) | = 2(x^2 - 8x) + 2(y^2 - 2y) | ||
= 2(x^2 - 8x + 16) - (2)(16) + 2(y^2 - 2y + 1) - (2)(1) | = 2(x^2 - 8x + 16) - (2)(16) + 2(y^2 - 2y + 1) - (2)(1) | ||
= 2(x - 4)^2 + 2(y - 1)^2 - 34 | = 2(x - 4)^2 + 2(y - 1)^2 - 34 | ||
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All squared values must be greater or equal to 0. As we are aiming for the minimum value, we let the 2 squared terms be 0. This leads to (h+k)min = 0 + 0 - 34 = (C) -34 | All squared values must be greater or equal to 0. As we are aiming for the minimum value, we let the 2 squared terms be 0. This leads to (h+k)min = 0 + 0 - 34 = (C) -34 | ||
~mitsuihisashi14 | ~mitsuihisashi14 | ||
Revision as of 01:15, 14 November 2024
Solution 1: Easy
Adding up the first and second statement, we get h+k with:
= 2x^2 + 2y^2 - 16x - 4y = 2(x^2 - 8x) + 2(y^2 - 2y) = 2(x^2 - 8x + 16) - (2)(16) + 2(y^2 - 2y + 1) - (2)(1) = 2(x - 4)^2 + 2(y - 1)^2 - 34
All squared values must be greater or equal to 0. As we are aiming for the minimum value, we let the 2 squared terms be 0. This leads to (h+k)min = 0 + 0 - 34 = (C) -34
~mitsuihisashi14
