💡Okay, so I have this problem here about finding the maximum number of points where two different fifth-degree polynomial graphs can intersect. Both polynomials have a leading coefficient of 1. The options are from A to E, with the answers 2, 3, 4, 5, and 6. Hmm, let me think about how to approach this.First, I remember that when two graphs intersect, their y-values are equal at those points. So, if I have two polynomials p(x) and q(x), the points of intersection occur where p(x) = q(x). That means I can set them equal to each other and solve for x.So, p(x) = q(x). If I subtract q(x) from both sides, I get p(x) - q(x) = 0. Let me call this difference r(x) = p(x) - q(x). So, r(x) is another polynomial, right?Now, both p(x) and q(x) are fifth-degree polynomials with leading coefficient 1. So, their leading terms are x^5. When I subtract q(x) from p(x), the x^5 terms will cancel out because they're both 1x^5. That means the highest degree term in r(x) will be the next highest degree term, which is x^4. So, r(x) is a fourth-degree polynomial, also known as a quartic polynomial.I remember from the Fundamental Theorem of Algebra that a polynomial of degree n can have at most n real roots. So, since r(x) is a fourth-degree polynomial, it can have at most 4 real roots. Each real root corresponds to a point where p(x) and q(x) intersect. Therefore, the maximum number of intersection points is 4.Wait, let me make sure I'm not missing something. Could there be a case where r(x) has more than 4 roots? Well, no, because the degree is 4, so it can't have more than 4 roots. Even if some roots are repeated, the maximum number of distinct intersection points is still 4.Is there a possibility that the difference polynomial r(x) could be of a higher degree? No, because both p(x) and q(x) are fifth-degree, and their leading terms cancel out, leaving a fourth-degree polynomial. So, r(x) can't be higher than fourth-degree.Therefore, the maximum number of points where the two graphs can intersect is 4. Looking back at the options, that's choice C.**Final Answer**boxed{C}

💡Okay, so I need to figure out how many interior diagonals a dodecahedron has. First, I should probably recall what a dodecahedron is. From the problem statement, it's a 3D shape with 12 pentagonal faces and 20 vertices. Each vertex is where three faces meet. An interior diagonal is defined as a segment connecting two vertices that don't lie on the same face. Alright, so to find the number of interior diagonals, I think I need to consider all possible line segments between vertices and then subtract those that are edges or face diagonals. But wait, in a dodecahedron, each face is a pentagon, so there are no diagonals on the faces in the traditional sense because all vertices on a face are connected by edges. Hmm, actually, in a pentagon, there are diagonals, right? So maybe I need to consider those as well.Let me break it down step by step.First, the total number of line segments connecting any two vertices in a dodecahedron can be calculated using combinations. Since there are 20 vertices, the total number of line segments is C(20, 2), which is 190. Now, out of these 190 line segments, some are the edges of the dodecahedron, and the rest are diagonals. But not all diagonals are interior; some are face diagonals. So I need to subtract both the edges and the face diagonals from the total to get the number of interior diagonals.First, let's find the number of edges. In a dodecahedron, each of the 12 faces is a pentagon, and each pentagon has 5 edges. However, each edge is shared by two faces. So the total number of edges is (12 * 5) / 2 = 30 edges.Next, let's find the number of face diagonals. Since each face is a pentagon, the number of diagonals in a pentagon is given by C(5, 2) - 5 = 5 diagonals per face. So for 12 faces, that's 12 * 5 = 60 face diagonals.Now, subtracting the edges and face diagonals from the total number of line segments should give me the number of interior diagonals. So that's 190 (total) - 30 (edges) - 60 (face diagonals) = 100 interior diagonals.Wait, but I'm not sure if this is correct. Let me double-check. Each face has 5 diagonals, and there are 12 faces, so 60 face diagonals. The total number of line segments is 190, and edges are 30. So 190 - 30 - 60 = 100. That seems right.But I recall that in some polyhedrons, especially the regular ones like the dodecahedron, there might be a different way to count interior diagonals. Maybe I should consider the number of diagonals per vertex and then multiply by the number of vertices, adjusting for overcounting.Each vertex in a dodecahedron is connected to 3 other vertices by edges (since three faces meet at each vertex). So, from each vertex, there are 20 - 1 - 3 = 16 possible diagonals. But wait, not all of these are interior. Some of these diagonals lie on the faces, which we've already considered. So, actually, from each vertex, the number of interior diagonals would be 16 - number of face diagonals from that vertex.But how many face diagonals come out of each vertex? Each vertex is part of three faces, and in each face, a vertex is connected to two adjacent vertices by edges, and the other two vertices are connected by diagonals. So, in each face, from a given vertex, there are 2 face diagonals. Since there are three faces meeting at each vertex, that's 3 * 2 = 6 face diagonals per vertex.Therefore, the number of interior diagonals per vertex would be 16 - 6 = 10. So, each vertex has 10 interior diagonals.Now, if I multiply this by the number of vertices, 20 * 10 = 200. But wait, this counts each interior diagonal twice because each diagonal connects two vertices. So, to get the actual number of unique interior diagonals, I need to divide by 2. So, 200 / 2 = 100.Okay, so both methods give me 100 interior diagonals. That seems consistent. I think my initial calculation was correct.But just to be thorough, let me check another approach. The formula for the number of space diagonals in a convex polyhedron is given by:Number of space diagonals = (V * (V - 3)) / 2 - EWhere V is the number of vertices and E is the number of edges.Plugging in the numbers, V = 20, E = 30.So, (20 * (20 - 3)) / 2 - 30 = (20 * 17) / 2 - 30 = (340) / 2 - 30 = 170 - 30 = 140.Wait, that's different. Now I'm getting 140. Hmm, that's conflicting with my previous result of 100.I must have made a mistake somewhere. Let me see. The formula (V * (V - 3)) / 2 gives the total number of diagonals in the polyhedron, including both face diagonals and space diagonals. Then subtracting the number of edges gives the total number of diagonals, both face and space. But I want only the space diagonals, which are the interior ones.Wait, no. Actually, the formula (V * (V - 3)) / 2 gives the total number of diagonals in the polyhedron, which includes both face diagonals and space diagonals. So to get only the space diagonals, I need to subtract the face diagonals from this total.So, total diagonals = (20 * 17) / 2 = 170.Face diagonals = 60 (as calculated earlier).Therefore, space diagonals = 170 - 60 = 110.Wait, now I'm getting 110. This is different from both previous results.I'm confused now. Let me try to clarify.First, the total number of line segments between vertices is C(20, 2) = 190.Out of these, 30 are edges.The remaining 190 - 30 = 160 are diagonals, both face and space.Now, out of these 160 diagonals, 60 are face diagonals, so the remaining should be space diagonals: 160 - 60 = 100.So, that brings me back to 100.But when I used the formula (V * (V - 3)) / 2 - E, I got 140, which seems incorrect because it should be 170 - 30 = 140, but that counts both face and space diagonals. Wait, no, actually, the formula (V * (V - 3)) / 2 is the total number of diagonals, which is 170, and subtracting edges gives 170 - 30 = 140, but that's not correct because edges are not diagonals. Wait, no, the formula is supposed to give the number of diagonals, not including edges.Wait, let me check the formula again. The formula for the number of diagonals in a polyhedron is indeed (V * (V - 3)) / 2. This counts all diagonals, both face and space. So, in our case, that's 170.But we have 60 face diagonals, so the number of space diagonals should be 170 - 60 = 110.But earlier, by subtracting edges and face diagonals from the total line segments, I got 100.There's a discrepancy here. Which one is correct?Wait, maybe I made a mistake in counting face diagonals. Let me recalculate the number of face diagonals.Each face is a pentagon, which has C(5, 2) - 5 = 5 diagonals. So, 12 faces * 5 diagonals = 60 face diagonals. That seems correct.Total line segments: 190.Edges: 30.Face diagonals: 60.So, space diagonals: 190 - 30 - 60 = 100.But according to the formula, total diagonals (face + space) = 170, so space diagonals = 170 - 60 = 110.But 190 - 30 - 60 = 100.So, which is correct?Wait, perhaps the formula (V * (V - 3)) / 2 counts all diagonals, but in a polyhedron, some of these diagonals are face diagonals, and the rest are space diagonals.But in our case, the total number of diagonals is 170, which includes both face and space diagonals.But when we subtract edges, we get 170, which is the total number of diagonals. So, to get space diagonals, we need to subtract face diagonals from this.So, space diagonals = 170 - 60 = 110.But earlier, by subtracting edges and face diagonals from total line segments, I got 100.This inconsistency is concerning. I need to figure out where I went wrong.Wait, perhaps the formula (V * (V - 3)) / 2 is not applicable to polyhedrons in the same way as it is to polygons.In a polygon, the number of diagonals is C(n, 2) - n = n(n - 3)/2.But in a polyhedron, the concept is similar but more complex because diagonals can be face diagonals or space diagonals.Wait, actually, in a polyhedron, the total number of diagonals (both face and space) is indeed C(V, 2) - E, which is 190 - 30 = 160.So, that contradicts the formula (V * (V - 3)) / 2, which gave 170.Wait, so which one is correct?Let me calculate C(20, 2) - E = 190 - 30 = 160. So, total diagonals (face + space) = 160.But according to the formula (V * (V - 3)) / 2, it's 20 * 17 / 2 = 170.So, there's a discrepancy of 10.Hmm, that suggests that the formula (V * (V - 3)) / 2 is not applicable to polyhedrons in the same way as polygons.Perhaps in polyhedrons, the formula is different because not all pairs of vertices are connected by edges or face diagonals.Wait, in a polygon, every pair of non-adjacent vertices is connected by a diagonal. But in a polyhedron, some pairs of vertices are not on the same face, so their connecting line is a space diagonal.Therefore, the total number of diagonals in a polyhedron is C(V, 2) - E, which is 190 - 30 = 160.Out of these 160, 60 are face diagonals, so the remaining 100 are space diagonals.Therefore, the correct number of interior diagonals (space diagonals) is 100.So, why did the formula (V * (V - 3)) / 2 give 170? Maybe that formula is not applicable to polyhedrons, or perhaps it's a different concept.Wait, actually, in graph theory, the number of edges in a complete graph is C(V, 2). In a polyhedron, the number of edges is E, so the number of non-edges is C(V, 2) - E, which are the diagonals. So, in this case, 190 - 30 = 160 diagonals.Out of these, 60 are face diagonals, so 160 - 60 = 100 space diagonals.Therefore, the correct number is 100.I think the confusion came from misapplying the formula from polygons to polyhedrons. The formula (V * (V - 3)) / 2 is for polygons, not polyhedrons.So, to summarize:Total line segments: C(20, 2) = 190.Edges: 30.Face diagonals: 60.Space diagonals (interior diagonals): 190 - 30 - 60 = 100.Therefore, the number of interior diagonals in a dodecahedron is 100.

💡Okay, so I have this equation here: -x² - (k + 10)x - b = -(x - 2)(x - 4), and I know that b is 8. I need to find the value of k. Hmm, let's see. I think I should start by expanding the right side of the equation to make it easier to compare with the left side.Alright, so the right side is -(x - 2)(x - 4). I remember that when you multiply two binomials, you can use the FOIL method: First, Outer, Inner, Last. Let me try that.First, I'll multiply the first terms: x * x = x².Next, the Outer terms: x * (-4) = -4x.Then, the Inner terms: (-2) * x = -2x.Lastly, the Last terms: (-2) * (-4) = 8.So, putting it all together, (x - 2)(x - 4) = x² - 4x - 2x + 8. Wait, that simplifies to x² - 6x + 8. But there's a negative sign in front, so it becomes -x² + 6x - 8.Okay, so now the equation looks like this: -x² - (k + 10)x - 8 = -x² + 6x - 8.Hmm, let's see. Both sides have -x² and -8, so they cancel out. That leaves me with -(k + 10)x = 6x.Wait, so -(k + 10)x equals 6x. To find k, I can set the coefficients equal to each other. So, -(k + 10) should equal 6.Let me write that down: -(k + 10) = 6.Now, I need to solve for k. If I distribute the negative sign, I get -k - 10 = 6.Then, I'll add 10 to both sides: -k = 6 + 10, which is 16.So, -k = 16. To find k, I'll multiply both sides by -1, which gives me k = -16.Wait, does that make sense? Let me check my steps again to make sure I didn't make a mistake.First, expanding -(x - 2)(x - 4) gave me -x² + 6x - 8. Then, setting that equal to the left side: -x² - (k + 10)x - 8. Comparing the coefficients of x, I had -(k + 10) = 6, leading to k = -16.Yeah, that seems right. I think k is -16.

💡Okay, so I have this problem with vectors and points, and I need to solve two parts. Let me try to understand what each part is asking.First, the problem gives me points A, B, and C with their position vectors from the origin O. The vectors are:- Vector OA = (3, -4)- Vector OB = (6, -3)- Vector OC = (5 - m, -3 - m)So, points A, B, and C are defined by these vectors. Now, part (1) says that points A, B, and C are collinear, and I need to find the real number m. Hmm, collinear means that all three points lie on the same straight line. So, I need to figure out the value of m that makes this happen.Part (2) is a bit different. It says that triangle ABC is a right triangle with the right angle at point A. So, I need to find m such that when I connect points A, B, and C, the triangle formed has a right angle at A.Alright, let's tackle part (1) first.**Problem (1): Collinear Points**I remember that for three points to be collinear, the vectors between them must be scalar multiples of each other. That is, the vector from A to B and the vector from A to C should be parallel. So, if I can express vector AB as a scalar multiple of vector AC, then points A, B, and C are collinear.First, let me find vectors AB and AC.Vector AB is found by subtracting vector OA from vector OB:AB = OB - OA = (6, -3) - (3, -4) = (6 - 3, -3 - (-4)) = (3, 1)Similarly, vector AC is found by subtracting vector OA from vector OC:AC = OC - OA = (5 - m, -3 - m) - (3, -4) = (5 - m - 3, -3 - m - (-4)) = (2 - m, 1 - m)So, vector AB is (3, 1) and vector AC is (2 - m, 1 - m).Since AB and AC must be parallel, there exists some scalar k such that AB = k * AC.So, component-wise, this gives:3 = k * (2 - m) ...(1)1 = k * (1 - m) ...(2)Now, I have two equations with two unknowns: k and m. I can solve for k from both equations and then set them equal to each other.From equation (1):k = 3 / (2 - m)From equation (2):k = 1 / (1 - m)So, setting them equal:3 / (2 - m) = 1 / (1 - m)Cross-multiplying:3 * (1 - m) = 1 * (2 - m)Expanding both sides:3 - 3m = 2 - mNow, let's solve for m:3 - 3m = 2 - mSubtract 2 from both sides:1 - 3m = -mAdd 3m to both sides:1 = 2mDivide both sides by 2:m = 1/2So, m is 1/2. Let me just check my steps to make sure I didn't make a mistake.1. Calculated AB and AC correctly.2. Set AB = k * AC, leading to two equations.3. Solved for k in both equations and set them equal.4. Cross-multiplied and solved for m, getting m = 1/2.Seems solid. So, part (1) answer is m = 1/2.**Problem (2): Right Triangle at A**Now, part (2) says that triangle ABC is a right triangle with the right angle at A. So, the vectors AB and AC should be perpendicular to each other because the right angle is at A.I remember that two vectors are perpendicular if their dot product is zero. So, the dot product of AB and AC should be zero.First, let me recall vectors AB and AC:AB = (3, 1)AC = (2 - m, 1 - m)So, the dot product AB · AC is:(3)(2 - m) + (1)(1 - m) = 0Let me compute this:3*(2 - m) + 1*(1 - m) = 0Expanding:6 - 3m + 1 - m = 0Combine like terms:(6 + 1) + (-3m - m) = 07 - 4m = 0Now, solve for m:-4m = -7Divide both sides by -4:m = 7/4So, m is 7/4. Let me double-check my steps.1. Calculated vectors AB and AC correctly.2. Set up the dot product equation.3. Expanded and combined like terms.4. Solved for m, getting 7/4.Looks good. So, part (2) answer is m = 7/4.**Summary**So, after working through both parts, I found that:1. For points A, B, and C to be collinear, m must be 1/2.2. For triangle ABC to be a right triangle with the right angle at A, m must be 7/4.I think I did everything correctly, but just to be thorough, let me plug m = 1/2 back into vector AC for part (1) and see if AB is a scalar multiple.Vector AC when m = 1/2:(2 - 1/2, 1 - 1/2) = (3/2, 1/2)So, vector AC is (3/2, 1/2). Vector AB is (3, 1). Indeed, AB is 2 times AC because 3 = 2*(3/2) and 1 = 2*(1/2). So, that checks out.For part (2), plugging m = 7/4 into vector AC:(2 - 7/4, 1 - 7/4) = (1/4, -3/4)So, vector AC is (1/4, -3/4). Vector AB is (3, 1). Let's compute the dot product:3*(1/4) + 1*(-3/4) = 3/4 - 3/4 = 0Which confirms that they are perpendicular. So, that also checks out.Therefore, I'm confident that my answers are correct.**Final Answer**(1) boxed{dfrac{1}{2}} (2) boxed{dfrac{7}{4}}

💡To find the greatest common divisor (GCD) of 360 and 504, I will use the prime factorization method.First, I will factorize each number into its prime components.For 360:360 can be divided by 2: 360 ÷ 2 = 180180 can be divided by 2: 180 ÷ 2 = 9090 can be divided by 2: 90 ÷ 2 = 4545 can be divided by 3: 45 ÷ 3 = 1515 can be divided by 3: 15 ÷ 3 = 55 is a prime number.So, the prime factors of 360 are 2 × 2 × 2 × 3 × 3 × 5.For 504:504 can be divided by 2: 504 ÷ 2 = 252252 can be divided by 2: 252 ÷ 2 = 126126 can be divided by 2: 126 ÷ 2 = 6363 can be divided by 3: 63 ÷ 3 = 2121 can be divided by 3: 21 ÷ 3 = 77 is a prime number.So, the prime factors of 504 are 2 × 2 × 2 × 3 × 3 × 7.Next, I will identify the common prime factors between the two numbers.Both 360 and 504 have three 2s and two 3s in their prime factorizations.Multiplying these common prime factors together gives the GCD:2 × 2 × 2 × 3 × 3 = 72Therefore, the greatest common divisor of 360 and 504 is 72.

💡Okay, so I have this problem where ABCD is a square, and points E and F are on sides BC and CD respectively. The triangle AEF is a right triangle with AE = 4 and EF = 3. I need to find the area of the square ABCD.First, I should probably draw a diagram to visualize the problem. Since ABCD is a square, all sides are equal, and all angles are right angles. Points E and F are somewhere on BC and CD, respectively. Triangle AEF is a right triangle, so one of its angles is 90 degrees. The problem doesn't specify which angle is the right angle, but since AE and EF are given, it's likely that the right angle is at E or F. I'll assume it's at E for now, but I should check if that makes sense.Let me label the square ABCD with A at the top-left corner, B at the top-right, C at the bottom-right, and D at the bottom-left. So, side AB is the top, BC is the right side, CD is the bottom, and DA is the left side. Point E is somewhere on BC, and point F is somewhere on CD.Since AEF is a right triangle with AE = 4 and EF = 3, I need to figure out where E and F are located on BC and CD. Let me assign some variables to the lengths. Let's say the side length of the square is 's'. Then, AB = BC = CD = DA = s.Let me denote BE as x, so EC would be s - x. Similarly, let me denote CF as y, so FD would be s - y.Now, triangle AEF is a right triangle. If the right angle is at E, then AE and EF are the legs, and AF would be the hypotenuse. But wait, AF is not necessarily a side of the square; it's a diagonal from A to F. Alternatively, if the right angle is at F, then AF and EF would be the legs, and AE would be the hypotenuse. But AE is given as 4, which is longer than EF = 3, so it's possible that the right angle is at E.Wait, actually, in a right triangle, the hypotenuse is the longest side. So if AE = 4 and EF = 3, then AF must be the hypotenuse if the right angle is at E. Alternatively, if the right angle is at F, then AE would be the hypotenuse, which is longer than EF, so that's possible too. Hmm, I need to clarify this.Let me consider both cases:1. Right angle at E: Then, AE and EF are the legs, so AF is the hypotenuse. Then, by Pythagoras, AF² = AE² + EF² = 4² + 3² = 16 + 9 = 25, so AF = 5.2. Right angle at F: Then, AF and EF are the legs, and AE is the hypotenuse. So, AE² = AF² + EF². But AE is 4, so 4² = AF² + 3² => 16 = AF² + 9 => AF² = 7 => AF = √7. But AF is a diagonal from A to F, which is inside the square, so its length should be less than the diagonal of the square, which is s√2. So, AF = √7 is possible if s√2 > √7, which would mean s > √(7/2) ≈ 1.87. But we don't know s yet.However, if the right angle is at E, AF = 5, which is a more straightforward number, so maybe that's the intended case. I'll proceed with the assumption that the right angle is at E, so AF = 5.Now, I need to express AF in terms of the square's side length 's'. Let's see, point F is on CD, so its coordinates can be expressed in terms of s and y (where y is CF). Similarly, point E is on BC, so its coordinates can be expressed in terms of s and x (where x is BE).Let me assign coordinates to the square to make it easier. Let's place point A at (0, s), B at (s, s), C at (s, 0), and D at (0, 0). Then, point E is on BC, so its coordinates are (s, s - x), since BE = x. Similarly, point F is on CD, so its coordinates are (s - y, 0), since CF = y.Now, point A is at (0, s), E is at (s, s - x), and F is at (s - y, 0). The triangle AEF has vertices at these three points.Since we're assuming the right angle is at E, the vectors AE and EF should be perpendicular. Let's compute the vectors:Vector AE is from A to E: (s - 0, (s - x) - s) = (s, -x).Vector EF is from E to F: ((s - y) - s, 0 - (s - x)) = (-y, -(s - x)).For these vectors to be perpendicular, their dot product should be zero:(s)(-y) + (-x)(-(s - x)) = 0=> -sy + x(s - x) = 0=> -sy + sx - x² = 0=> s(x - y) = x²So, s = x² / (x - y)Hmm, that's one equation.Also, we know that AF = 5. Let's compute AF using coordinates. Point A is at (0, s), and point F is at (s - y, 0). So, the distance AF is:AF² = (s - y - 0)² + (0 - s)² = (s - y)² + s² = 25So,(s - y)² + s² = 25Expanding (s - y)²:s² - 2sy + y² + s² = 25=> 2s² - 2sy + y² = 25Now, from earlier, we have s = x² / (x - y). Let's substitute s into this equation.First, let's express y in terms of x and s from s = x² / (x - y):s(x - y) = x²=> sx - sy = x²=> sy = sx - x²=> y = (sx - x²)/s = x - x²/sSo, y = x - x²/sNow, substitute y into the equation 2s² - 2sy + y² = 25.First, compute each term:2s² is straightforward.-2sy: we have y = x - x²/s, so -2sy = -2s(x - x²/s) = -2sx + 2x²y²: (x - x²/s)² = x² - 2x³/s + x⁴/s²So, putting it all together:2s² - 2sy + y² = 2s² - 2sx + 2x² + x² - 2x³/s + x⁴/s² = 25Simplify:2s² - 2sx + 2x² + x² - 2x³/s + x⁴/s² = 25Combine like terms:2s² - 2sx + 3x² - 2x³/s + x⁴/s² = 25This looks complicated. Maybe there's a better approach.Alternatively, since triangle AEF is a right triangle at E, we can use coordinates to find the lengths.We have AE = 4, EF = 3, and AF = 5.From coordinates:AE is the distance from A(0, s) to E(s, s - x):AE² = (s - 0)² + ((s - x) - s)² = s² + (-x)² = s² + x² = 16So,s² + x² = 16 ...(1)Similarly, EF is the distance from E(s, s - x) to F(s - y, 0):EF² = ((s - y) - s)² + (0 - (s - x))² = (-y)² + (-s + x)² = y² + (x - s)² = 9So,y² + (x - s)² = 9 ...(2)And AF is the distance from A(0, s) to F(s - y, 0):AF² = (s - y - 0)² + (0 - s)² = (s - y)² + s² = 25So,(s - y)² + s² = 25 ...(3)Now, we have three equations:1. s² + x² = 162. y² + (x - s)² = 93. (s - y)² + s² = 25Let me try to solve these equations step by step.From equation (1): s² + x² = 16 => x² = 16 - s² ...(1a)From equation (3): (s - y)² + s² = 25Expand (s - y)²:s² - 2sy + y² + s² = 25=> 2s² - 2sy + y² = 25 ...(3a)From equation (2): y² + (x - s)² = 9Expand (x - s)²:x² - 2sx + s² + y² = 9But from (1a), x² = 16 - s², so substitute:(16 - s²) - 2sx + s² + y² = 9Simplify:16 - s² - 2sx + s² + y² = 9=> 16 - 2sx + y² = 9=> y² = 2sx - 7 ...(2a)Now, substitute y² from (2a) into (3a):2s² - 2sy + (2sx - 7) = 25Simplify:2s² - 2sy + 2sx - 7 = 25=> 2s² + 2sx - 2sy = 32Divide both sides by 2:s² + sx - sy = 16 ...(4)Now, from equation (1a): x² = 16 - s²From equation (2a): y² = 2sx - 7Also, from equation (4): s² + sx - sy = 16Let me try to express y in terms of x and s from equation (4):s² + sx - sy = 16=> sy = s² + sx - 16=> y = (s² + sx - 16)/s = s + x - 16/s ...(4a)Now, substitute y from (4a) into equation (2a):y² = 2sx - 7=> (s + x - 16/s)² = 2sx - 7Expand the left side:(s + x - 16/s)² = s² + 2s(x) + x² - 2*(s)*(16/s) - 2*(x)*(16/s) + (16/s)²Wait, that's a bit messy. Let me do it step by step.Let me denote z = s + x - 16/s, so z² = (s + x - 16/s)²= (s + x)² - 2*(s + x)*(16/s) + (16/s)²= s² + 2sx + x² - 32 - (32x)/s + 256/s²So,z² = s² + 2sx + x² - 32 - (32x)/s + 256/s²Set this equal to 2sx - 7:s² + 2sx + x² - 32 - (32x)/s + 256/s² = 2sx - 7Subtract 2sx from both sides:s² + x² - 32 - (32x)/s + 256/s² = -7Bring all terms to the left:s² + x² - 32 - (32x)/s + 256/s² + 7 = 0Simplify:s² + x² - 25 - (32x)/s + 256/s² = 0 ...(5)Now, from equation (1a): x² = 16 - s²Substitute x² into equation (5):s² + (16 - s²) - 25 - (32x)/s + 256/s² = 0Simplify:s² + 16 - s² - 25 - (32x)/s + 256/s² = 0=> 16 - 25 - (32x)/s + 256/s² = 0=> -9 - (32x)/s + 256/s² = 0Multiply both sides by s² to eliminate denominators:-9s² - 32x s + 256 = 0=> 9s² + 32x s - 256 = 0 ...(6)Now, from equation (1a): x² = 16 - s² => x = sqrt(16 - s²). But we need to express x in terms of s to substitute into equation (6).Alternatively, from equation (4a): y = s + x - 16/sBut we also have from equation (2a): y² = 2sx - 7So, substituting y from (4a) into (2a):(s + x - 16/s)² = 2sx - 7Which is what we did earlier, leading to equation (5). So, perhaps another approach is needed.Let me consider equation (6): 9s² + 32x s - 256 = 0We can express x from equation (1a): x = sqrt(16 - s²)But that would complicate things with square roots. Alternatively, maybe we can express x in terms of s from another equation.Wait, from equation (4a): y = s + x - 16/sAnd from equation (2a): y² = 2sx - 7So, substituting y:(s + x - 16/s)² = 2sx - 7Which is equation (5). So, perhaps we need to find a relationship between s and x.Alternatively, let's consider that from equation (1a): x² = 16 - s²So, x = sqrt(16 - s²)Let me substitute x into equation (6):9s² + 32*sqrt(16 - s²)*s - 256 = 0This is a nonlinear equation in s. It might be challenging to solve algebraically, so perhaps we can make a substitution.Let me set t = s². Then, sqrt(16 - s²) = sqrt(16 - t)So, equation (6) becomes:9t + 32*sqrt(16 - t)*sqrt(t) - 256 = 0Wait, that's still complicated. Alternatively, let me denote u = s, so equation (6) is:9u² + 32u*sqrt(16 - u²) - 256 = 0This is a transcendental equation and might not have an algebraic solution. Maybe I can try to solve it numerically or look for rational solutions.Alternatively, perhaps there's a better way to approach the problem without coordinates.Let me think differently. Since ABCD is a square, and E is on BC, F is on CD, and AEF is a right triangle with AE = 4, EF = 3.Let me denote the side length of the square as s.Let me consider triangles ABE and EFC. Since ABCD is a square, angle at B is 90 degrees, and angle at E in triangle AEF is also 90 degrees. So, maybe triangles ABE and EFC are similar.Wait, if angle at E is 90 degrees, then triangle ABE is similar to triangle EFC because both are right triangles and they share angle at B and C respectively.So, similarity ratio would be AB/EC = AE/EF = 4/3So, AB = s, EC = s - BELet me denote BE = x, so EC = s - xFrom similarity, AB/EC = AE/EF = 4/3So,s / (s - x) = 4/3Cross-multiplying:3s = 4(s - x)=> 3s = 4s - 4x=> 4x = s=> x = s/4So, BE = s/4, EC = s - s/4 = 3s/4Now, from triangle ABE, which is a right triangle with legs AB = s and BE = s/4, and hypotenuse AE = 4.Wait, no, triangle ABE is a right triangle with legs AB and BE, but AB is the side of the square, which is s, and BE is s/4. So, by Pythagoras:AB² + BE² = AE²=> s² + (s/4)² = 4²=> s² + s²/16 = 16Multiply both sides by 16:16s² + s² = 256=> 17s² = 256=> s² = 256/17So, the area of the square is s² = 256/17Wait, that seems straightforward. Did I make a mistake earlier? Because this approach seems much simpler.Let me verify:Triangles ABE and EFC are similar because both are right triangles and they share angle at B and C respectively. So, the ratio of their corresponding sides is equal.AB corresponds to EC, and AE corresponds to EF.So, AB/EC = AE/EF = 4/3Thus, s / (s - x) = 4/3Solving for x gives x = s/4Then, in triangle ABE, which is a right triangle with legs AB = s and BE = s/4, hypotenuse AE = 4.So,s² + (s/4)² = 16=> s² + s²/16 = 16=> (16s² + s²)/16 = 16=> 17s² = 256=> s² = 256/17Yes, that's correct. So, the area of the square is 256/17.I think this is the correct approach. Earlier, I was complicating things with coordinates, but using similar triangles simplifies the problem significantly.