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question:What is the sum of all values of x that are solutions to the equation x^2 = 9x 20?
answer:First, we need to rearrange the given equation into standard quadratic form: [ x^2 - 9x + 20 = 0 ] Now, we apply the sum of the roots formula for a quadratic equation ax^2 + bx + c = 0, where the sum of the roots is given by -frac{b}{a}. Here, a = 1 and b = -9: [ text{Sum of roots} = -frac{-9}{1} = 9 ] Thus, the sum of all solutions to the equation x^2 = 9x - 20 is boxed{9}.
question:f f(5) = 3 and f(3x) = 3f(x) for all x, find f^{1}(243).
answer:We need to find x such that f(x) = 243. Given the function's property f(3x) = 3f(x) and f(5) = 3, we can apply the property repeatedly. [ f(5) = 3 ] [ f(15) = 9 ] [ f(45) = 27 ] [ f(135) = 81 ] [ f(405) = 243 ] Thus, f^{-1}(243) = boxed{405}.
question:The sum of all positive divisors of 120 that are composite numbers but not odd equals to ____.
answer:The positive divisors of 120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, and 120, totaling 16 divisors. Among them, the divisors that are composite but not odd are 4, 6, 8, 10, 12, 20, 24, 30, 40, 60, and 120, totaling 11 divisors. Their sum is 4 + 6 + 8 + 10 + 12 + 20 + 24 + 30 + 40 + 60 + 120 = 334. Therefore, the answer is boxed{334}.
question:When Jane visited Iran, she visited some more rehabilitation centers than twice as many as Han. Han visited 2 less than twice as many rehabilitation centers as Jude did. Jude visited half fewer rehabilitation centers as Lisa did. Lisa visited 6 rehabilitation centers. They all visited 27 different rehabilitation centers. How many more rehabilitation centers did Jane visit than twice as many as Han?
answer:Let's break down the information given step by step: 1. Lisa visited 6 rehabilitation centers. 2. Jude visited half fewer rehabilitation centers than Lisa, so Jude visited ( 6 / 2 = 3 ) rehabilitation centers. 3. Han visited 2 less than twice as many rehabilitation centers as Jude, so Han visited ( 2 * 3 - 2 = 6 - 2 = 4 ) rehabilitation centers. 4. Jane visited some more rehabilitation centers than twice as many as Han, so Jane visited ( 2 * 4 + x ) rehabilitation centers, where ( x ) is the number of additional centers Jane visited beyond twice the number Han visited. Now, we know that they all visited 27 different rehabilitation centers in total. So we can set up the following equation: [ 6 (Lisa) + 3 (Jude) + 4 (Han) + (8 + x) (Jane) = 27 ] Simplifying the equation: [ 6 + 3 + 4 + 8 + x = 27 ] [ 21 + x = 27 ] [ x = 27 - 21 ] [ x = 6 ] So, Jane visited 6 more rehabilitation centers than twice as many as Han. To find out how many rehabilitation centers Jane visited in total, we add twice the number Han visited (which is 8) to the additional centers Jane visited (which is 6): [ 8 + 6 = 14 ] Jane visited boxed{14} rehabilitation centers in total.