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question:Let C be the circle with equation x^2+12y+57=-y^2-10x. If (a,b) is the center of C and r is its radius, what is the value of a+b+r?
answer:We rewrite the equation as x^2 + 10x + y^2 + 12y = -57 and then complete the square, resulting in (x + 5)^2 - 25 + (y + 6)^2 - 36 = -57, or (x + 5)^2 + (y + 6)^2 = 4. This is the equation of a circle with center (-5, -6) and radius 2, so a = -5, b = -6, and r = 2. Therefore, a+b+r = -5 + (-6) + 2 = boxed{-9}.The answer is: -9
question:A pet shop has 2 puppies and some kittens. A puppy costs 20, and a kitten costs 15. If the stock is worth 100, how many kittens does the pet shop have?
answer:Let's assume the number of kittens is K. The total cost of the puppies is 2 * 20 = 40. The total cost of the kittens is K * 15 = 15K. The total stock is worth 100, so we have the equation 40 + 15K = 100. By subtracting 40 from both sides, we get 15K = 60. Dividing both sides by 15, we get K = 4. Therefore, the pet shop has 4 kittens. 4 The answer is: 4
question:Zack has traveled to twice the number of countries Patrick traveled to. Patrick traveled to three times the number of countries Joseph traveled to. Joseph traveled to half the number of countries George traveled to. If George traveled to x countries, how many countries did Zack travel to? If we know the answer to the above question is 18, what is the value of unknown variable x?
answer:We are given that George traveled to x countries. Joseph traveled to half the number of countries George traveled to, which means Joseph traveled to 0.5x countries. Patrick traveled to three times the number of countries Joseph traveled to, which means Patrick traveled to 3 * 0.5x = 1.5x countries. Zack has traveled to twice the number of countries Patrick traveled to, which means Zack traveled to 2 * 1.5x = 3x countries. We are given that Zack traveled to 18 countries, so we can write: 3x = 18. Solving for x, we get: x = 6. The value of x is 6. 6 The answer is: 6
question:Given that -1 - 4sqrt{2} is a root of the equation [x^3 + ax^2 + bx + 31 = 0]and that a and b are rational numbers, compute a.
answer:of the equation r_1, r_2, and -1 - 4sqrt{2}. By Vieta's formulas, we know that [r_1 + r_2 + (-1 - 4sqrt{2}) = -a]and that [(r_1)(r_2)(-1 - 4sqrt{2}) = -31.]Since a is rational, r_1 + r_2 must be rational, so r_1 + r_2 = -a - (-1 - 4sqrt{2}) = -a + 1 + 4sqrt{2}. Thus, [(-a + 1 + 4sqrt{2}) + (-1 - 4sqrt{2}) = -a,]so -2a = -2, which means a = boxed{1}. The answer is: 1