answer:4x^2 + x - 1 = 3 for the x-coordinates of the points of intersection. First, we move all terms to one side to get 4x^2 + x - 4 = 0. Next, we can factor this quadratic as (4x - 1)(x + 4) = 0. Setting each factor equal to zero, we have 4x - 1 = 0 and x + 4 = 0. Solving these equations, we find x = frac{1}{4} and x = -4. So, the points of intersection are left(frac{1}{4}, 3right) and (-4, 3). The distance between these points is sqrt{left(frac{1}{4} - (-4)right)^2 + (3 - 3)^2} = sqrt{frac{65}{16}} = frac{sqrt{65}}{4}. Therefore, m - n = 65 - 4 = boxed{61}.The answer is: 61

answer:Cindy gave her four friends 80 marbles each, so she gave away a total of 4 * 80 = 320 marbles. The number of marbles she has remaining is x - 320. We are given that four times the number of marbles she has remaining is 720, so we can write: 4 * (x - 320) = 720. Simplifying the left side, we get: 4x - 1280 = 720. Adding 1280 to both sides, we get: 4x = 2000. Dividing both sides by 4, we get: x = 500. The value of x is 500. 500 The answer is: 500

answer:We begin by expanding the right side of the equation to get x^2+13x. Next, we subtract x^2+13x from both sides to get 2x^2+24x-60-x^2-13x=0. Simplifying, we have x^2+11x-60=0. Factoring the quadratic, we get (x+15)(x-4)=0. Setting each factor equal to zero, we have x+15=0 or x-4=0. Solving each equation, we find x=-15 or x=4. The smallest possible value of x is boxed{-15}. The answer is: -15

answer:Machine A can cap 12 bottles in 1 minute, so in 10 minutes it can cap 12 x 10 = 120 bottles. Machine B can cap 2 fewer bottles than Machine A, so it can cap 12 - 2 = 10 bottles in 1 minute. In 10 minutes, Machine B can cap 10 x 10 = 100 bottles. Machine C can cap 5 more bottles than Machine B, so it can cap 10 + 5 = 15 bottles in 1 minute. In 10 minutes, Machine C can cap 15 x 10 = 150 bottles. Together, these three machines can cap 120 + 100 + 150 = 370 bottles in 10 minutes. 370 The answer is: 370