answer：Okay, so I have this equation here: ( x^2 - 10x - 1 = 0 ). I need to transform it into one of the given forms, which are all perfect squares set equal to some number. The options are:A: ( (x - 5)^2 = 26 )B: ( (x + 5)^2 = 26 )C: ( (x - 5)^2 = 24 )D: ( (x + 5)^2 = 24 )Hmm, I think this involves completing the square. I remember that completing the square is a method used to solve quadratic equations by turning them into a perfect square trinomial, which can then be easily solved.Alright, let's start with the original equation:( x^2 - 10x - 1 = 0 )First, I want to get rid of the constant term on the left side so I can focus on completing the square with the quadratic and linear terms. To do that, I'll add 1 to both sides of the equation:( x^2 - 10x = 1 )Now, the equation is ( x^2 - 10x = 1 ). The next step in completing the square is to take the coefficient of the x-term, which is -10, divide it by 2, and then square it. So, let's do that:Coefficient of x: -10Divide by 2: -10 / 2 = -5Square it: (-5)^2 = 25So, I need to add 25 to both sides of the equation to complete the square:( x^2 - 10x + 25 = 1 + 25 )Simplifying the right side:( x^2 - 10x + 25 = 26 )Now, the left side is a perfect square trinomial. Let's write it as a squared binomial:( (x - 5)^2 = 26 )So, the transformed equation is ( (x - 5)^2 = 26 ), which matches option A.Wait, let me double-check to make sure I didn't make any mistakes. Starting from the original equation:( x^2 - 10x - 1 = 0 )Adding 1 to both sides:( x^2 - 10x = 1 )Completing the square by adding 25:( x^2 - 10x + 25 = 26 )Which becomes:( (x - 5)^2 = 26 )Yes, that seems correct. So, the correct transformation is option A.I wonder why the other options are there. Let me quickly check what would happen if I made some mistakes in the process.If I incorrectly added a different number instead of 25, say 24, then the equation would be:( (x - 5)^2 = 24 ), which is option C. But I know that to complete the square correctly, I needed to add 25, not 24, because (-5)^2 is 25.What if I had added 25 to the wrong side? That would have given me 1 + 25 = 26 on the right, which is correct. If I had subtracted instead, that would have been wrong, but I didn't do that.Also, if I had mistakenly used +5 instead of -5 in the binomial, that would have given me ( (x + 5)^2 ), which is options B and D. But since the original coefficient was -10, half of that is -5, so it should be ( (x - 5)^2 ).So, all the other options are incorrect because they either have the wrong constant term after completing the square or the wrong sign in the binomial.Therefore, I'm confident that option A is the correct transformation.

answer：Okay, so I have this problem where I need to determine which pair of functions are "accessible functions." The definition given is that two function graphs are considered accessible if the distance between them is less than 1. There are four options: A, B, C, and D. I need to check each pair and see if their distance is less than 1.Starting with option A: f(x) = cos x and g(x) = 2. Hmm, cos x oscillates between -1 and 1. So the maximum value of cos x is 1, and the minimum is -1. The function g(x) is a constant function at 2. So the distance between these two functions would be the vertical distance between them. The closest point would be when cos x is at its maximum, which is 1. So the distance would be |2 - 1| = 1. But the requirement is that the distance is less than 1, so 1 doesn't satisfy the condition. Therefore, option A is not accessible.Moving on to option B: f(x) = log₂(x² - 2x + 5) and g(x) = sin(π/2 x). Let me analyze f(x) first. The expression inside the logarithm is x² - 2x + 5. Completing the square, that's (x - 1)² + 4. So the minimum value of the inside function is 4 when x = 1. Therefore, f(x) has a minimum value of log₂(4) = 2. The function g(x) is a sine function with amplitude 1, so it oscillates between -1 and 1. The maximum value of g(x) is 1. So the distance between f(x) and g(x) would be the vertical distance between them. The closest they get is when f(x) is at its minimum (2) and g(x) is at its maximum (1). So the distance is |2 - 1| = 1. Again, it's equal to 1, not less than 1. So option B is also not accessible.Next, option C: f(x) = sqrt(4 - x²) and g(x) = (3/4)x + 15/4. Let's see. The function f(x) is the upper half of a circle with radius 2 centered at the origin because sqrt(4 - x²) is the equation of a semicircle. The function g(x) is a straight line. To find the distance between the circle and the line, I can calculate the distance from the center of the circle (0,0) to the line and then subtract the radius. The formula for the distance from a point (x₀, y₀) to the line ax + by + c = 0 is |ax₀ + by₀ + c| / sqrt(a² + b²). Let me rewrite g(x) in standard form: (3/4)x - y + 15/4 = 0. So a = 3/4, b = -1, c = 15/4. Plugging in (0,0): |0 + 0 + 15/4| / sqrt((3/4)² + (-1)²) = (15/4) / sqrt(9/16 + 1) = (15/4) / sqrt(25/16) = (15/4) / (5/4) = 3. So the distance from the center to the line is 3. Since the radius is 2, the distance between the circle and the line is 3 - 2 = 1. Again, equal to 1, so it doesn't satisfy the less than 1 condition. Therefore, option C is not accessible.Finally, option D: f(x) = x + 2/x and g(x) = ln x + 2. To find the distance between these two functions, I need to consider the vertical distance between them, which is |f(x) - g(x)|. Let me define h(x) = f(x) - g(x) = x + 2/x - ln x - 2. I need to find the minimum value of |h(x)|. Since h(x) is a continuous function for x > 0 (because of the ln x and 2/x terms), I can find its critical points by taking the derivative.Calculating h'(x): derivative of x is 1, derivative of 2/x is -2/x², derivative of -ln x is -1/x, and derivative of -2 is 0. So h'(x) = 1 - 2/x² - 1/x. Let me combine these terms: h'(x) = (x² - 2 - x)/x². Wait, that might not be the best way. Alternatively, factor it differently. Let me write h'(x) as 1 - 1/x - 2/x². To find critical points, set h'(x) = 0: 1 - 1/x - 2/x² = 0. Multiply both sides by x² to eliminate denominators: x² - x - 2 = 0. Solving this quadratic equation: x = [1 ± sqrt(1 + 8)] / 2 = [1 ± 3]/2. So x = 2 or x = -1. Since x must be positive, we discard x = -1. So the critical point is at x = 2.Now, let's check the behavior of h(x) around x = 2. For x < 2, say x = 1, h'(1) = 1 - 1 - 2 = -2 < 0. For x > 2, say x = 3, h'(3) = 1 - 1/3 - 2/9 = 1 - 1/3 - 2/9 = (9/9 - 3/9 - 2/9) = 4/9 > 0. So h(x) is decreasing before x = 2 and increasing after x = 2, meaning x = 2 is a minimum point.Calculating h(2): 2 + 2/2 - ln 2 - 2 = 2 + 1 - ln 2 - 2 = 1 - ln 2. Since ln 2 is approximately 0.693, 1 - 0.693 ≈ 0.307, which is less than 1. Therefore, the minimum distance between f(x) and g(x) is approximately 0.307, which is less than 1. So option D satisfies the condition.So after checking all options, only option D meets the requirement of having a distance less than 1 between the two functions.

answer：First, I need to understand what a proper subset is. A proper subset of a set ( A ) is any subset of ( A ) that is not equal to ( A ) itself.Given the universal set ( A = {0, 1, 2} ), the total number of subsets of ( A ) can be calculated using the formula ( 2^n ), where ( n ) is the number of elements in the set. Since ( A ) has 3 elements, the total number of subsets is ( 2^3 = 8 ).To find the number of proper subsets, I subtract the set ( A ) itself from the total number of subsets. Therefore, the number of proper subsets is ( 8 - 1 = 7 ).Listing all the proper subsets of ( A ):- The empty set: ( emptyset )- Single-element subsets: ( {0} ), ( {1} ), ( {2} )- Two-element subsets: ( {0, 1} ), ( {0, 2} ), ( {1, 2} )This confirms that there are 7 proper subsets of ( A ).

answer：Okay, so I have this problem where there's a square, and inside the square, there's a point P. The distances from P to three of the square's vertices, A, B, and C, are given as a, b, and c respectively. I need to find the area of the square, and specifically, what happens when a, b, and c are all equal.Hmm, let me visualize this. Imagine a square ABCD, with A, B, C, D as the vertices in order. Point P is somewhere inside this square. The distances from P to A, B, and C are a, b, and c. I need to find the area of the square, which would be the side length squared.Since it's a square, all sides are equal, and all angles are right angles. Maybe I can use coordinate geometry here. Let me assign coordinates to the square. Let's say point A is at (0, 0), B is at (s, 0), C is at (s, s), and D is at (0, s), where s is the side length of the square. Then, point P has coordinates (x, y) inside the square.So, the distance from P to A is a, which would be sqrt((x)^2 + (y)^2) = a.Similarly, the distance from P to B is b, which is sqrt((x - s)^2 + (y)^2) = b.And the distance from P to C is c, which is sqrt((x - s)^2 + (y - s)^2) = c.So, I have three equations:1. x² + y² = a²2. (x - s)² + y² = b²3. (x - s)² + (y - s)² = c²I need to solve these equations to find s, the side length of the square. Then, the area would be s².Let me see if I can subtract equation 1 from equation 2 to eliminate y².Equation 2 minus equation 1: (x - s)² + y² - (x² + y²) = b² - a²Expanding (x - s)²: x² - 2s x + s² + y² - x² - y² = b² - a²Simplify: -2s x + s² = b² - a²So, -2s x = b² - a² - s²Therefore, x = (s² + a² - b²) / (2s)Okay, so I have x expressed in terms of s, a, and b.Similarly, maybe I can subtract equation 2 from equation 3 to get another equation.Equation 3 minus equation 2: (x - s)² + (y - s)² - [(x - s)² + y²] = c² - b²Simplify: (y - s)² - y² = c² - b²Expanding (y - s)²: y² - 2s y + s² - y² = c² - b²Simplify: -2s y + s² = c² - b²So, -2s y = c² - b² - s²Therefore, y = (s² + b² - c²) / (2s)Alright, so now I have both x and y in terms of s, a, b, and c.Now, since I have expressions for x and y, I can plug these back into equation 1, which is x² + y² = a².Let me write that out:[(s² + a² - b²)/(2s)]² + [(s² + b² - c²)/(2s)]² = a²Let me compute each term:First term: [(s² + a² - b²)/(2s)]² = (s² + a² - b²)² / (4s²)Second term: [(s² + b² - c²)/(2s)]² = (s² + b² - c²)² / (4s²)So, adding them together:[(s² + a² - b²)² + (s² + b² - c²)²] / (4s²) = a²Multiply both sides by 4s²:(s² + a² - b²)² + (s² + b² - c²)² = 4s² a²Hmm, this seems a bit complicated, but maybe I can expand the left side.Let me denote A = s² + a² - b² and B = s² + b² - c².Then, the equation becomes A² + B² = 4s² a².Expanding A² and B²:A² = (s² + a² - b²)² = s⁴ + 2s²(a² - b²) + (a² - b²)²Similarly, B² = (s² + b² - c²)² = s⁴ + 2s²(b² - c²) + (b² - c²)²Adding A² and B²:A² + B² = 2s⁴ + 2s²(a² - b² + b² - c²) + (a² - b²)² + (b² - c²)²Simplify the middle term:2s²(a² - c²)So, A² + B² = 2s⁴ + 2s²(a² - c²) + (a² - b²)² + (b² - c²)²Set this equal to 4s² a²:2s⁴ + 2s²(a² - c²) + (a² - b²)² + (b² - c²)² = 4s² a²Let me bring all terms to one side:2s⁴ + 2s²(a² - c²) + (a² - b²)² + (b² - c²)² - 4s² a² = 0Combine like terms:2s⁴ + 2s²(a² - c² - 2a²) + (a² - b²)² + (b² - c²)² = 0Simplify the coefficient of s²:a² - c² - 2a² = -a² - c²So:2s⁴ - 2s²(a² + c²) + (a² - b²)² + (b² - c²)² = 0Let me write this as:2s⁴ - 2(a² + c²)s² + [(a² - b²)² + (b² - c²)²] = 0This is a quadratic equation in terms of s². Let me let u = s².Then, the equation becomes:2u² - 2(a² + c²)u + [(a² - b²)² + (b² - c²)²] = 0Divide both sides by 2 to simplify:u² - (a² + c²)u + [(a² - b²)² + (b² - c²)²]/2 = 0Now, let me compute the discriminant of this quadratic equation to solve for u.The discriminant D is:D = [-(a² + c²)]² - 4 * 1 * [(a² - b²)² + (b² - c²)²]/2Simplify:D = (a² + c²)² - 2[(a² - b²)² + (b² - c²)²]Let me expand (a² + c²)²:(a² + c²)² = a⁴ + 2a²c² + c⁴Now, expand (a² - b²)² and (b² - c²)²:(a² - b²)² = a⁴ - 2a²b² + b⁴(b² - c²)² = b⁴ - 2b²c² + c⁴So, adding them together:(a² - b²)² + (b² - c²)² = a⁴ - 2a²b² + b⁴ + b⁴ - 2b²c² + c⁴ = a⁴ + 2b⁴ + c⁴ - 2a²b² - 2b²c²Multiply by 2:2[(a² - b²)² + (b² - c²)²] = 2a⁴ + 4b⁴ + 2c⁴ - 4a²b² - 4b²c²Now, compute D:D = (a⁴ + 2a²c² + c⁴) - (2a⁴ + 4b⁴ + 2c⁴ - 4a²b² - 4b²c²)Simplify term by term:a⁴ - 2a⁴ = -a⁴2a²c² remainsc⁴ - 2c⁴ = -c⁴Then, subtract the rest:-4b⁴ + 4a²b² + 4b²c²So, putting it all together:D = -a⁴ + 2a²c² - c⁴ - 4b⁴ + 4a²b² + 4b²c²Hmm, this seems messy. Maybe I can factor this expression.Let me rearrange terms:D = (-a⁴ - c⁴) + 2a²c² + (-4b⁴) + 4a²b² + 4b²c²Notice that -a⁴ + 2a²c² - c⁴ = -(a⁴ - 2a²c² + c⁴) = -(a² - c²)²Similarly, the other terms: -4b⁴ + 4a²b² + 4b²c² = -4b⁴ + 4b²(a² + c²)So, D = -(a² - c²)² -4b⁴ + 4b²(a² + c²)Let me factor out -4b⁴ + 4b²(a² + c²):= -4b⁴ + 4b²(a² + c²) = 4b²(a² + c² - b²) -4b⁴Wait, maybe factor 4b²:= 4b²(a² + c² - b²) -4b⁴ = 4b²(a² + c² - b² - b²) = 4b²(a² + c² - 2b²)So, putting it all together:D = -(a² - c²)² + 4b²(a² + c² - 2b²)Hmm, not sure if that helps. Maybe another approach.Alternatively, let me factor D differently.Wait, perhaps I made a mistake in expanding. Let me double-check.Original D:D = (a² + c²)² - 2[(a² - b²)² + (b² - c²)²]Compute each part:(a² + c²)² = a⁴ + 2a²c² + c⁴2[(a² - b²)² + (b² - c²)²] = 2(a⁴ - 2a²b² + b⁴ + b⁴ - 2b²c² + c⁴) = 2a⁴ - 4a²b² + 2b⁴ + 2b⁴ - 4b²c² + 2c⁴ = 2a⁴ + 4b⁴ + 2c⁴ - 4a²b² - 4b²c²So, D = (a⁴ + 2a²c² + c⁴) - (2a⁴ + 4b⁴ + 2c⁴ - 4a²b² - 4b²c²)= a⁴ + 2a²c² + c⁴ - 2a⁴ - 4b⁴ - 2c⁴ + 4a²b² + 4b²c²= -a⁴ - c⁴ - 4b⁴ + 2a²c² + 4a²b² + 4b²c²Hmm, same as before.Maybe factor terms:= (-a⁴ + 4a²b² - 4b⁴) + (2a²c² + 4b²c² - c⁴)Factor first group:- (a⁴ - 4a²b² + 4b⁴) = - (a² - 2b²)²Second group:2a²c² + 4b²c² - c⁴ = c²(2a² + 4b² - c²)So, D = - (a² - 2b²)² + c²(2a² + 4b² - c²)Hmm, not sure if that helps.Alternatively, maybe I can write D as:D = 4a²c² - (2b² - a² - c²)²Wait, let me check:Let me compute (2b² - a² - c²)²:= (2b² - a² - c²)² = 4b⁴ + a⁴ + c⁴ - 4a²b² - 4b²c² + 2a²c²So, 4a²c² - (2b² - a² - c²)² = 4a²c² - [4b⁴ + a⁴ + c⁴ - 4a²b² - 4b²c² + 2a²c²] = 4a²c² -4b⁴ -a⁴ -c⁴ +4a²b² +4b²c² -2a²c² = (-a⁴ - c⁴ -4b⁴) + (4a²b² +4b²c² +2a²c²)Which is exactly D. So, D = 4a²c² - (2b² - a² - c²)²That's a nice expression.So, D = 4a²c² - (2b² - a² - c²)²Therefore, the discriminant is D = [2b² - (a² + c²)]² - 4a²c², but wait, actually, D = 4a²c² - (2b² - a² - c²)²So, D = 4a²c² - (2b² - a² - c²)²Hmm, interesting. So, for the quadratic equation in u, which is s², the solutions are:u = [ (a² + c²) ± sqrt(D) ] / 2So, s² = [ (a² + c²) ± sqrt(4a²c² - (2b² - a² - c²)²) ] / 2Hmm, that looks complicated, but maybe it can be simplified.Let me compute sqrt(D):sqrt(D) = sqrt(4a²c² - (2b² - a² - c²)²)Let me factor this expression:Let me denote X = 2b² - a² - c²Then, D = 4a²c² - X²So, sqrt(D) = sqrt(4a²c² - X²) = sqrt( (2ac - X)(2ac + X) )But X = 2b² - a² - c², so:= sqrt( (2ac - 2b² + a² + c²)(2ac + 2b² - a² - c²) )Hmm, maybe factor further:First term: 2ac - 2b² + a² + c² = a² + 2ac + c² - 2b² = (a + c)² - 2b²Second term: 2ac + 2b² - a² - c² = -a² + 2ac - c² + 2b² = -(a² - 2ac + c²) + 2b² = -(a - c)² + 2b²So, sqrt(D) = sqrt( [ (a + c)² - 2b² ] [ 2b² - (a - c)² ] )Therefore, sqrt(D) = sqrt( ( (a + c)² - 2b² )( 2b² - (a - c)² ) )So, putting it back into the expression for s²:s² = [ (a² + c²) ± sqrt( ( (a + c)² - 2b² )( 2b² - (a - c)² ) ) ] / 2Hmm, this is quite involved. I wonder if there's a condition for the discriminant to be non-negative, which would require that the product inside the square root is non-negative.So, for sqrt(D) to be real, we need:( (a + c)² - 2b² )( 2b² - (a - c)² ) ≥ 0Which implies that either both factors are non-negative or both are non-positive.Case 1: Both factors are non-negative.So,(a + c)² - 2b² ≥ 0 and 2b² - (a - c)² ≥ 0Which implies:(a + c)² ≥ 2b² and (a - c)² ≤ 2b²Similarly, Case 2: Both factors are non-positive.(a + c)² - 2b² ≤ 0 and 2b² - (a - c)² ≤ 0Which implies:(a + c)² ≤ 2b² and (a - c)² ≥ 2b²But since (a + c)² is always greater than or equal to (a - c)², because (a + c)² - (a - c)² = 4ac ≥ 0, the second case would require (a + c)² ≤ 2b² and (a - c)² ≥ 2b², but since (a + c)² ≥ (a - c)², this would imply (a + c)² ≤ 2b² and (a - c)² ≥ 2b², which would require (a + c)² ≤ 2b² and (a - c)² ≥ 2b², but since (a + c)² ≥ (a - c)², this would require 2b² ≤ (a - c)² ≤ (a + c)² ≤ 2b², which would imply (a - c)² = (a + c)² = 2b², which is only possible if ac = 0, but since a, b, c are distances, they are positive, so ac ≠ 0. Therefore, Case 2 is not possible.Therefore, only Case 1 is valid, which requires:(a + c)² ≥ 2b² and (a - c)² ≤ 2b²So, these are the conditions for real solutions.Now, going back to the expression for s²:s² = [ (a² + c²) ± sqrt( ( (a + c)² - 2b² )( 2b² - (a - c)² ) ) ] / 2I need to decide which sign to take. Since s² must be positive, and the expression inside the square root is non-negative, both solutions might be positive, but we need to choose the one that makes sense geometrically.Let me consider the case when a = b = c. Maybe that will help me check if my formula is correct.So, if a = b = c, then let's substitute into the expression.First, compute s²:s² = [ (a² + a²) ± sqrt( ( (a + a)² - 2a² )( 2a² - (a - a)² ) ) ] / 2Simplify:= [ 2a² ± sqrt( (4a² - 2a²)(2a² - 0) ) ] / 2= [ 2a² ± sqrt( (2a²)(2a²) ) ] / 2= [ 2a² ± sqrt(4a⁴) ] / 2= [ 2a² ± 2a² ] / 2So, two possibilities:1. [2a² + 2a²]/2 = 4a²/2 = 2a²2. [2a² - 2a²]/2 = 0/2 = 0But s² can't be zero, so we take the first solution: s² = 2a²Therefore, when a = b = c, the area of the square is 2a².That makes sense because if all distances from P to A, B, and C are equal, P must be the center of the square. In a square, the distance from the center to any vertex is (s√2)/2, so setting that equal to a:(s√2)/2 = a => s = a√2 => s² = 2a²So, that checks out.But what about the general case? Is there a simpler expression for s²?Looking back at the expression:s² = [ (a² + c²) ± sqrt( ( (a + c)² - 2b² )( 2b² - (a - c)² ) ) ] / 2It's quite complex, but maybe it can be written in terms of the sides of a triangle or something.Alternatively, perhaps there's a different approach to the problem.Wait, another thought: in a square, the distances from an interior point to three consecutive vertices satisfy certain relations. Maybe using the British flag theorem? Wait, the British flag theorem states that for any point P in the plane of a rectangle, the sum of the squares of the distances from P to two opposite corners equals the sum of the squares of the distances to the other two opposite corners.But in this case, we have distances to three consecutive vertices, not two opposite ones. So, maybe not directly applicable.But let me recall the British flag theorem:For rectangle ABCD, PA² + PC² = PB² + PD²But in our case, we have PA = a, PB = b, PC = c, and PD is unknown. So, PD² = PA² + PC² - PB² = a² + c² - b²So, PD = sqrt(a² + c² - b²)But I don't know if that helps directly, but maybe.Wait, but in our earlier equations, we had expressions for x and y in terms of s, a, b, c.We had:x = (s² + a² - b²)/(2s)y = (s² + b² - c²)/(2s)And then plugging into equation 1: x² + y² = a²Which led us to the quadratic in s².Alternatively, maybe I can use the Pythagorean theorem in some way.Wait, another idea: since ABCD is a square, the coordinates of P can be used to form right triangles with the sides of the square.But I think I'm going in circles here. Maybe I should accept that the expression for s² is as complicated as it is, and that in the general case, it's given by that quadratic solution.But perhaps there's a different approach using coordinate geometry or vectors.Alternatively, maybe using complex numbers.But perhaps it's better to stick with the result we have.So, in general, the area of the square is s², which is given by:s² = [ (a² + c²) + sqrt( ( (a + c)² - 2b² )( 2b² - (a - c)² ) ) ] / 2And when a = b = c, s² = 2a².Therefore, the area of the square is 2a² when a = b = c.I think that's the answer.**Final Answer**The area of the square is boxed{2a^2} when (a = b = c).

answer：First, I need to find the first term of the arithmetic sequence. I know that the eighth term is 20 and the common difference is 2. Using the formula for the nth term of an arithmetic sequence, which is ( a_n = a + (n-1)d ), where ( a ) is the first term and ( d ) is the common difference, I can set up the equation for the eighth term:[a + 7d = 20]Substituting ( d = 2 ):[a + 7 cdot 2 = 20 implies a + 14 = 20 implies a = 6]Now that I have the first term, I can find the third and fourth terms. The third term is calculated as:[a + 2d = 6 + 2 cdot 2 = 6 + 4 = 10]The fourth term is:[a + 3d = 6 + 3 cdot 2 = 6 + 6 = 12]Finally, I need to find the product of the third and fourth terms:[10 cdot 12 = 120]

answer：First, I need to determine how many blue marbles Sandy has. Since Sandy has 8 times more blue marbles than Melanie, I will multiply Melanie's number of blue marbles by 8.Sandy's blue marbles = Melanie's blue marbles * 8Sandy's blue marbles = 84 * 8Sandy's blue marbles = 672Next, to find out how many dozen blue marbles Sandy has, I will divide the total number of blue marbles by 12, as one dozen equals 12.Sandy's dozen blue marbles = Sandy's blue marbles / 12Sandy's dozen blue marbles = 672 / 12Sandy's dozen blue marbles = 56Therefore, Sandy has 56 dozen blue marbles.