Solve the following for x by completing the square: (x^2)+4x-60 = 0. Assume the domain for x is (0, infinity).
Adding 60 to both sides, we are left with (x^2)+4x = 60. We can complete the square of (x+2) by adding 4 to both sides, which gives us (x^2)+4x+4 = 64. The left side of the equation can be factored to (x+2)^2, thus we are left with (x+2)^2 = 64. Taking the square root of both sides, we obtain (x+2) = +/- 8. Since x must be positive, we can subtract 2 from both sides leaving us with x = 6 as the final answer.
Lucia is a local village girl, who must, like all villagers, make a weekly trudge down to the magical water course to get water to fill her pail for the week. The water source is run by a magical fairy who charges the village for water based on the time of the week. While the rest of the village can't figure out the method of her prices, Lucia gets in contact with some friends, and, using her math skills, comes to the conclusion that the prices relate roughly to the function 3(x^2)-15x+25, where x represents the day of the week. 0 = Sunday, 1 = Monday..., and 6 = Saturday. Assuming her assumptions are correct, If Lucia wants to minimize the cost she pays for water, what day of the week should she fill her pail?
We are looking to minimize the function Y = 3(X^2) - 15X + 25 for X to determine the absolute lowest price we can pay for the water. To find the minimum, we must differentiate the function and solve X for zero. Differentiating, we get 6X-15 = 0. Solving for X, we obtain X = 15/6, which is 2.5. This may be the minimum value of the function, but to be certain, we must also check the endpoint, 0 and 6. Plugging 2.5, 0, and 6 into the initial equation and solving we get 6.25, 25, and 67, respectively. Thus, we conclude that the X value which gives us the lowest cost is 2.5. Is Sunday is 0, Monday is 1, then Tuesday must be 2. Thus, in order to get the lowest price, Lucia must fill her pail at exactly between Tuesday and Wednesday.
Kevin has $16 to buy snacks for his friends for a party he's throwing tonight. He knows his four invitees like Doritos and the rest of his friends like Lay's. If Doritos cost $2 a bag and Lay's cost $3 and assuming no taxes or additional costs, what is the maximum number of additional friends he can invite to his party to ensure everyone gets their own bag of chips?
The previous question can be answered by setting up and solving the linear equation: 2X + 3Y <= 16 where the 2 represents the cost of the Doritos, 3 represents the cost of Lay's, and X & Y represent the number of bags he can buy of Doritos and Lay's, respectively. The question tells us that X = 4, so multiplying that by 2 and subtracting the result from 16, we are left with 4Y <= 10. Diving both sides by 4, we get Y<=10/4, which is 2.5. Since Y must be less than or equal to 2.5 and we can only buy full bags of chips, Y must be equal to 2. Therefore, Kevin can only invite 2 more people to his party.