answer：First, I need to determine the total number of ways to form a five-person subcommittee from the 12 members of the planning committee. This can be calculated using the combination formula ( {12 choose 5} ).Next, I'll calculate the number of possible subcommittees that consist entirely of non-teachers. Since there are 7 non-teachers, the number of ways to choose 5 non-teachers is ( {7 choose 5} ).Finally, to find the number of subcommittees that include at least one teacher, I'll subtract the number of all-non-teacher subcommittees from the total number of possible subcommittees. This will give me the desired number of valid subcommittees.

answer：First, I need to understand the relationship between the two numbers, x and y. The problem states that x is 6 times y, which can be written as x = 6y.Next, I want to determine what percentage y is less than x. To do this, I'll calculate the difference between x and y, which is x - y. Substituting x with 6y, the difference becomes 6y - y = 5y.Now, to find out what percentage this difference represents relative to x, I'll divide the difference by x and then multiply by 100 to convert it to a percentage. So, the calculation is (5y / 6y) * 100.Simplifying this, the y terms cancel out, leaving (5/6) * 100, which equals approximately 83.33%.Therefore, y is 83.33% less than x.

answer：Okay, so I have this problem about a tetrahedron ABCD and a point P inside or on the surface of it, but P isn't the same as vertex D. I need to prove that one of the segments PA, PB, or PC is shorter than one of DA, DB, or DC. Hmm, tetrahedrons are like 3D shapes with four triangular faces, right? So, ABCD is a tetrahedron, and P is somewhere inside or on the faces, but not at D.First, I should visualize this. Maybe drawing a rough sketch would help. Let me imagine tetrahedron ABCD with vertices A, B, C, and D. Point P is somewhere inside or on the faces, but not at D. So, P could be on face ABC, or inside the tetrahedron.The problem is asking to show that at least one of PA, PB, or PC is shorter than at least one of DA, DB, or DC. So, for example, maybe PA is shorter than DA, or PB is shorter than DB, or PC is shorter than DC. At least one of these has to be true.I think I should consider the positions of P relative to D. Since P isn't D, it's somewhere else. If P is on one of the edges connected to D, like DA, DB, or DC, then it's straightforward. For example, if P is on DA, then PA is just part of DA, so PA would be shorter than DA. Similarly, if P is on DB or DC, then PB or PC would be shorter than DB or DC respectively. So, in that case, the statement is true.But what if P isn't on any of the edges connected to D? Then P is either inside the tetrahedron or on one of the other faces, like ABC. In that case, how can I relate PA, PB, PC to DA, DB, DC?Maybe I can use some geometric properties or inequalities. Perhaps considering the distances from P to the vertices and comparing them to the distances from D to the vertices. Since P is inside the tetrahedron, maybe there's a way to show that one of these distances has to be smaller.Wait, maybe I can think about the angles at point P. If I consider the angles between the segments PD and PA, PD and PB, PD and PC, maybe one of those angles is less than 90 degrees. If that's the case, then by the cosine law, the distance from P to A, B, or C would be less than the distance from D to A, B, or C.Let me try to formalize this. Suppose I look at triangle APD. If the angle at P, which is angle APD, is less than 90 degrees, then by the cosine law, PA² + PD² - 2*PA*PD*cos(angle APD) = AD². Since angle APD is less than 90 degrees, cos(angle APD) is positive, so PA² would be less than AD², meaning PA < AD.Similarly, if angle BPD or angle CPD is less than 90 degrees, then PB < BD or PC < CD respectively.But how do I know that at least one of these angles is less than 90 degrees? Well, the sum of the angles around point P in the plane of the tetrahedron is 360 degrees. If all three angles APD, BPD, and CPD were 90 degrees or more, their sum would be at least 270 degrees, leaving only 90 degrees for the remaining angles, which might not be possible. Wait, actually, in 3D space, the angles around P aren't necessarily in the same plane, so this might not hold.Hmm, maybe I need a different approach. Perhaps using vectors or coordinates. If I assign coordinates to the tetrahedron, I can express the distances PA, PB, PC in terms of coordinates and compare them to DA, DB, DC.Let me set up a coordinate system. Let’s place point D at the origin (0,0,0). Let’s assign coordinates to A, B, and C as vectors **a**, **b**, and **c** respectively. Then, point P can be represented as a convex combination of A, B, C, and D since it's inside or on the surface of the tetrahedron.So, P can be written as P = λA + μB + νC + τD, where λ, μ, ν, τ are non-negative and sum to 1. Since P is not D, at least one of λ, μ, ν is positive.Now, the distance from P to A is |P - A|, which is |(λA + μB + νC + τD) - A| = |(λ - 1)A + μB + νC + τD|. Similarly, distances to B and C can be expressed.But this seems complicated. Maybe instead, I can use the fact that in a tetrahedron, any interior point is closer to at least one vertex than the opposite vertex. Wait, is that a theorem? I'm not sure, but it sounds plausible.Alternatively, maybe I can use the pigeonhole principle. Since P is inside the tetrahedron, it can't be too far from all the vertices A, B, and C. So, at least one of PA, PB, or PC must be less than the corresponding DA, DB, or DC.Wait, but how do I formalize that? Maybe by considering the distances from P to A, B, C and comparing them to the distances from D to A, B, C.Let me think about the function f(x) = distance from x to A, and similarly for B and C. Since P is inside the tetrahedron, which is a convex set, the minimum distance from P to the vertices should be less than or equal to the distances from D to those vertices.But I'm not sure if that's the case. Maybe I need to use some inequality like the triangle inequality.Wait, the triangle inequality says that for any triangle, the sum of two sides is greater than the third. Maybe I can apply that here.Consider triangle APD. By the triangle inequality, PA + PD > AD. But that doesn't directly help me because I need PA < AD.Alternatively, maybe using the law of cosines. If I can show that the angle at P is acute, then PA < AD.But how do I ensure that at least one of the angles at P is acute? Maybe by considering the solid angles around P.Wait, in 3D space, the sum of the solid angles around a point is 4π steradians. But I'm not sure if that helps here.Maybe I should think about projections. If I project P onto the face ABC, then the distance from P to A, B, or C would be less than or equal to the distance from D to A, B, or C.Wait, that might make sense. Since D is opposite the face ABC, the distance from D to A, B, or C is the length of the edge DA, DB, or DC. If P is on the face ABC, then its distance to A, B, or C could be less than DA, DB, or DC.But P could be inside the tetrahedron, not necessarily on the face. So, maybe the projection idea still applies.Alternatively, maybe I can use the fact that in a tetrahedron, the distance from any interior point to a vertex is less than the distance from the opposite vertex to that vertex.Wait, that sounds like what I need. So, if P is inside the tetrahedron, then PA < DA, PB < DB, or PC < DC.But I need to prove that at least one of these is true. So, maybe assuming the contrary, that PA ≥ DA, PB ≥ DB, and PC ≥ DC, and reaching a contradiction.Let me try that. Suppose that PA ≥ DA, PB ≥ DB, and PC ≥ DC. Then, what can I say about the position of P?If PA ≥ DA, then P lies outside or on the sphere centered at A with radius DA. Similarly, P lies outside or on the spheres centered at B and C with radii DB and DC respectively.But since P is inside the tetrahedron ABCD, which is bounded by the faces, maybe this leads to a contradiction because P can't be outside all those spheres and still be inside the tetrahedron.Wait, but the spheres centered at A, B, C with radii DA, DB, DC might not necessarily cover the entire tetrahedron. So, maybe P can be inside the tetrahedron and still be outside all those spheres.Hmm, I'm not sure. Maybe I need a different approach.Let me think about the centroid of the tetrahedron. The centroid is the point where all the medians intersect, and it's equidistant from all vertices in some sense. But I don't know if that helps here.Alternatively, maybe I can use the fact that the tetrahedron can be divided into smaller tetrahedrons or regions where the distances from P to A, B, C are compared to DA, DB, DC.Wait, maybe considering the Voronoi regions around A, B, C with respect to D. The Voronoi region around A would consist of all points closer to A than to D. If P is in the Voronoi region of A, then PA < DA, and similarly for B and C.But since P is inside the tetrahedron, it must lie in at least one of the Voronoi regions of A, B, or C. Otherwise, it would be closer to D than to A, B, and C, which might not be possible inside the tetrahedron.Wait, that sounds promising. So, if P is inside the tetrahedron, it can't be closer to D than to all of A, B, and C. Therefore, P must be closer to at least one of A, B, or C than to D. Hence, PA < DA, PB < DB, or PC < DC.Yes, that makes sense. So, using the concept of Voronoi regions, I can argue that since P is inside the tetrahedron, it must lie within the Voronoi region of at least one of A, B, or C, meaning that the distance from P to that vertex is less than the distance from D to that vertex.Therefore, one of PA, PB, or PC is shorter than one of DA, DB, or DC.I think that's a solid argument. It uses the idea that within a tetrahedron, any interior point can't be equidistant or farther from all the vertices compared to the opposite vertex. So, at least one distance must be shorter.Another way to think about it is by considering the convex hull. Since P is inside the convex hull of A, B, C, D, it must be closer to at least one of the vertices A, B, or C than D is.Wait, actually, D is a vertex, so maybe it's not exactly the convex hull, but the tetrahedron itself. Still, the idea is similar.So, to summarize, by considering the Voronoi regions or the properties of points inside a tetrahedron, we can conclude that at least one of PA, PB, or PC must be shorter than the corresponding DA, DB, or DC.I think that's a good approach. It might be a bit abstract, but it uses fundamental geometric principles to reach the conclusion.

answer：Okay, so I have this problem here where I need to figure out the relationship between two statements, p and q. Let me read it again carefully.Statement p says that a, b, c, d form a geometric sequence. Statement q says that ad equals bc. The question is asking whether p is a sufficient condition, necessary condition, both, or neither for q. The options are A, B, C, D, corresponding to sufficient but not necessary, necessary but not sufficient, necessary and sufficient, or neither.Alright, so first, I need to recall what a geometric sequence is. In a geometric sequence, each term after the first is found by multiplying the previous term by a constant called the common ratio. So, if a, b, c, d are in a geometric sequence, then b = a * r, c = b * r = a * r^2, and d = c * r = a * r^3, where r is the common ratio.Given that, let's see what ad and bc would be. If I compute ad, that would be a * d. Since d = a * r^3, then ad = a * (a * r^3) = a^2 * r^3.Now, bc is b * c. Since b = a * r and c = a * r^2, then bc = (a * r) * (a * r^2) = a^2 * r^3.So, ad equals bc in this case. That means if p is true, then q must be true. Therefore, p is a sufficient condition for q.But wait, is p a necessary condition for q? In other words, does q imply p? To check this, I need to see if ad = bc necessarily means that a, b, c, d form a geometric sequence.Let me think of a counterexample where ad = bc, but a, b, c, d are not in a geometric sequence. For instance, suppose a = 1, b = 2, c = 3, d = 6. Let's compute ad and bc.ad = 1 * 6 = 6, and bc = 2 * 3 = 6. So, ad = bc. But are a, b, c, d in a geometric sequence? Let's check the ratios.b/a = 2/1 = 2, c/b = 3/2 = 1.5, d/c = 6/3 = 2. The ratios are not consistent (2, 1.5, 2), so it's not a geometric sequence. Therefore, q can hold without p holding, which means p is not a necessary condition for q.So, p is sufficient for q, but not necessary. Therefore, the correct answer should be A: Sufficient but not necessary condition.Wait, but let me test another example to make sure. Suppose a = 0, b = 0, c = 0, d = 0. Then ad = 0*0 = 0, and bc = 0*0 = 0. So, ad = bc. But in this case, a, b, c, d are all zero, which technically is a geometric sequence with any common ratio, but it's a trivial case. Maybe that's not the best example.Let me try another one where a, b, c, d are not all zero. Suppose a = 2, b = 4, c = 6, d = 12. Then ad = 2*12 = 24, and bc = 4*6 = 24. So, ad = bc. Are they in a geometric sequence? Let's check the ratios.b/a = 4/2 = 2, c/b = 6/4 = 1.5, d/c = 12/6 = 2. Again, the ratios are inconsistent. So, even though ad = bc, they don't form a geometric sequence. This reinforces that p is not necessary for q.Another thought: if a, b, c, d are in a geometric sequence, then ad = bc. But if ad = bc, does that necessarily mean they are in a geometric sequence? From the examples, it seems not. So, p is sufficient but not necessary.Wait, but could there be cases where ad = bc only if they are in a geometric sequence? Maybe if we have non-zero terms? Let me see. Suppose a, b, c, d are non-zero. Then, if ad = bc, does that imply that b/a = c/b = d/c?Let me write the equations:From ad = bc, we have a*d = b*c.If I rearrange, a/b = c/d.So, the ratio of the first to the second term is equal to the ratio of the third to the fourth term.But in a geometric sequence, the ratio of consecutive terms is constant, so b/a = c/b = d/c.So, if a/b = c/d, does that imply b/a = c/b?Wait, let's see:If a/b = c/d, then cross-multiplying gives a*d = b*c, which is the same as ad = bc.But does a/b = c/d imply b/a = c/b?Let me see:Suppose a/b = c/d. Let me denote a/b as k. Then, c/d = k as well.So, a = k*b and c = k*d.Now, let's compute b/a: b/a = 1/k.Similarly, c/b = (k*d)/b = k*(d/b). But from a = k*b, we have d = c/k.Wait, this is getting a bit tangled. Maybe I should express all terms in terms of a and k.If a/b = k, then b = a/k.Similarly, c = k*d, so d = c/k.But then, c can be expressed in terms of b, which is a/k. So, c = k*d, but d = c/k, which seems redundant.Wait, maybe I need to express all terms in terms of a and k.Let me try:Given a/b = c/d = k.So, b = a/k, c = k*d.But c is also the next term after b. If it were a geometric sequence, c should be b*r, where r is the common ratio.But here, c = k*d, and d is the term after c. So, if it were a geometric sequence, d should be c*r.But from c = k*d, we have d = c/k, which would imply that r = 1/k.So, if c = b*r, then c = (a/k)*r.But c is also equal to k*d, and d = c/k.So, substituting d into c = k*d, we get c = k*(c/k) = c. Which is just an identity, so it doesn't give us new information.Hmm, so maybe if a/b = c/d, it's not necessarily a geometric sequence unless the ratios are consistent throughout.So, in other words, if a/b = c/d, it doesn't automatically mean that b/a = c/b, unless certain conditions are met.Therefore, ad = bc doesn't necessarily lead to a geometric sequence unless the ratios between consecutive terms are equal.So, in conclusion, p implies q, but q does not necessarily imply p. Therefore, p is a sufficient condition for q, but not a necessary one.I think that solidifies my earlier conclusion. So, the answer should be A: Sufficient but not necessary condition.**Final Answer**boxed{A}

answer：Alright, so I have this problem where I need to prove that ( u = operatorname{ctg} frac{R}{4} ) satisfies a quadratic equation, and ( v = frac{1}{sin frac{R}{4}} ) satisfies a quartic equation with integer coefficients. The highest degree term in both equations should have a coefficient of 1. Here, ( R ) is 90 degrees. First, let me make sure I understand the problem correctly. I need to find equations for ( u ) and ( v ) such that when I plug in the values of ( u ) and ( v ) based on ( R = 90^circ ), the equations hold true, and the coefficients are integers with the leading coefficient being 1.Starting with ( u = operatorname{ctg} frac{R}{4} ). Since ( R = 90^circ ), ( frac{R}{4} = 22.5^circ ). So, ( u = operatorname{ctg} 22.5^circ ). I know that ( operatorname{ctg} theta = frac{1}{tan theta} ), so ( u = frac{1}{tan 22.5^circ} ).I remember that ( tan 22.5^circ ) can be expressed using the half-angle formula. The half-angle formula for tangent is ( tan frac{theta}{2} = frac{1 - cos theta}{sin theta} ). If I let ( theta = 45^circ ), then ( tan 22.5^circ = tan frac{45^circ}{2} = frac{1 - cos 45^circ}{sin 45^circ} ).Calculating ( cos 45^circ ) and ( sin 45^circ ), both are ( frac{sqrt{2}}{2} ). So, substituting these values in:( tan 22.5^circ = frac{1 - frac{sqrt{2}}{2}}{frac{sqrt{2}}{2}} = frac{frac{2 - sqrt{2}}{2}}{frac{sqrt{2}}{2}} = frac{2 - sqrt{2}}{sqrt{2}} ).Simplify this by rationalizing the denominator:( frac{2 - sqrt{2}}{sqrt{2}} times frac{sqrt{2}}{sqrt{2}} = frac{(2 - sqrt{2})sqrt{2}}{2} = frac{2sqrt{2} - 2}{2} = sqrt{2} - 1 ).So, ( tan 22.5^circ = sqrt{2} - 1 ), which means ( u = operatorname{ctg} 22.5^circ = frac{1}{sqrt{2} - 1} ).To rationalize the denominator:( frac{1}{sqrt{2} - 1} times frac{sqrt{2} + 1}{sqrt{2} + 1} = frac{sqrt{2} + 1}{(sqrt{2})^2 - 1^2} = frac{sqrt{2} + 1}{2 - 1} = sqrt{2} + 1 ).So, ( u = sqrt{2} + 1 ). Now, I need to find a quadratic equation that ( u ) satisfies. Let me set ( u = sqrt{2} + 1 ) and try to eliminate the square root.Let me subtract 1 from both sides:( u - 1 = sqrt{2} ).Now, square both sides:( (u - 1)^2 = (sqrt{2})^2 ).Expanding the left side:( u^2 - 2u + 1 = 2 ).Subtract 2 from both sides:( u^2 - 2u - 1 = 0 ).So, the quadratic equation is ( u^2 - 2u - 1 = 0 ). The coefficients are integers, and the leading coefficient is 1. That takes care of the first part.Now, moving on to ( v = frac{1}{sin frac{R}{4}} ). Again, ( R = 90^circ ), so ( frac{R}{4} = 22.5^circ ). Therefore, ( v = frac{1}{sin 22.5^circ} ).I need to find a quartic equation that ( v ) satisfies. Let me recall that ( sin 22.5^circ ) can be expressed using the half-angle formula as well. The half-angle formula for sine is ( sin frac{theta}{2} = sqrt{frac{1 - cos theta}{2}} ). Let me use ( theta = 45^circ ), so ( sin 22.5^circ = sin frac{45^circ}{2} = sqrt{frac{1 - cos 45^circ}{2}} ).Calculating ( cos 45^circ = frac{sqrt{2}}{2} ), so:( sin 22.5^circ = sqrt{frac{1 - frac{sqrt{2}}{2}}{2}} = sqrt{frac{frac{2 - sqrt{2}}{2}}{2}} = sqrt{frac{2 - sqrt{2}}{4}} = frac{sqrt{2 - sqrt{2}}}{2} ).Therefore, ( v = frac{1}{sin 22.5^circ} = frac{2}{sqrt{2 - sqrt{2}}} ).To rationalize the denominator, I can multiply numerator and denominator by ( sqrt{2 + sqrt{2}} ):( v = frac{2 sqrt{2 + sqrt{2}}}{sqrt{(2 - sqrt{2})(2 + sqrt{2})}} ).Calculating the denominator:( (2 - sqrt{2})(2 + sqrt{2}) = 4 - ( sqrt{2} )^2 = 4 - 2 = 2 ).So, ( v = frac{2 sqrt{2 + sqrt{2}}}{sqrt{2}} = frac{2}{sqrt{2}} sqrt{2 + sqrt{2}} = sqrt{2} sqrt{2 + sqrt{2}} ).Simplify further:( sqrt{2} times sqrt{2 + sqrt{2}} = sqrt{2(2 + sqrt{2})} = sqrt{4 + 2sqrt{2}} ).Hmm, this seems a bit complicated. Maybe there's a better way to approach this. Let me consider expressing ( v ) in terms of known quantities and then finding an equation it satisfies.Given that ( v = frac{1}{sin 22.5^circ} ), and I know that ( sin 22.5^circ = frac{sqrt{2 - sqrt{2}}}{2} ), so ( v = frac{2}{sqrt{2 - sqrt{2}}} ).Let me set ( v = frac{2}{sqrt{2 - sqrt{2}}} ) and try to find an equation for ( v ). Let me denote ( v = frac{2}{sqrt{2 - sqrt{2}}} ).Let me square both sides to eliminate the square root in the denominator:( v^2 = frac{4}{2 - sqrt{2}} ).Simplify the denominator by rationalizing:( v^2 = frac{4}{2 - sqrt{2}} times frac{2 + sqrt{2}}{2 + sqrt{2}} = frac{4(2 + sqrt{2})}{(2)^2 - (sqrt{2})^2} = frac{4(2 + sqrt{2})}{4 - 2} = frac{4(2 + sqrt{2})}{2} = 2(2 + sqrt{2}) = 4 + 2sqrt{2} ).So, ( v^2 = 4 + 2sqrt{2} ). Now, I need to eliminate the square root here. Let me isolate the square root term:( v^2 - 4 = 2sqrt{2} ).Divide both sides by 2:( frac{v^2 - 4}{2} = sqrt{2} ).Now, square both sides again:( left( frac{v^2 - 4}{2} right)^2 = (sqrt{2})^2 ).Simplify:( frac{(v^2 - 4)^2}{4} = 2 ).Multiply both sides by 4:( (v^2 - 4)^2 = 8 ).Expand the left side:( v^4 - 8v^2 + 16 = 8 ).Subtract 8 from both sides:( v^4 - 8v^2 + 8 = 0 ).So, the quartic equation is ( v^4 - 8v^2 + 8 = 0 ). The coefficients are integers, and the leading coefficient is 1. That completes the second part.Wait, let me double-check my steps to make sure I didn't make any mistakes. Starting from ( v = frac{1}{sin 22.5^circ} ), I expressed ( sin 22.5^circ ) using the half-angle formula, then rationalized and squared appropriately. The final quartic equation seems to check out.So, in summary, I found that ( u = sqrt{2} + 1 ) satisfies the quadratic equation ( u^2 - 2u - 1 = 0 ), and ( v = sqrt{4 + 2sqrt{2}} ) satisfies the quartic equation ( v^4 - 8v^2 + 8 = 0 ). Both equations have integer coefficients with leading coefficients of 1.I think I've covered all the necessary steps, but let me just recap to ensure clarity:1. For ( u ): - Expressed ( operatorname{ctg} 22.5^circ ) in terms of known trigonometric identities. - Rationalized and simplified to find ( u = sqrt{2} + 1 ). - Derived the quadratic equation by squaring and rearranging terms.2. For ( v ): - Expressed ( sin 22.5^circ ) using the half-angle formula. - Rationalized and squared appropriately to eliminate square roots. - Derived the quartic equation by isolating the square root and squaring again.Everything seems to align correctly, so I'm confident in the results.**Final Answer**The quadratic equation for ( u ) is ( boxed{u^2 - 2u - 1 = 0} ) and the quartic equation for ( v ) is ( boxed{v^4 - 8v^2 + 8 = 0} ).

answer：Okay, so I have this differential equation to solve: ( a frac{dy}{dx} + b left( frac{dy}{dx} right)^2 = x ). Hmm, it looks like a first-order equation, but it's not linear because of the squared term. Maybe I can simplify it by introducing a substitution. Let me think... If I let ( p = frac{dy}{dx} ), then the equation becomes ( a p + b p^2 = x ). That seems simpler. So now I have ( x = a p + b p^2 ). But I need to find ( y ) in terms of ( x ), right? So maybe I can express ( dx ) in terms of ( dp ). Let me try differentiating both sides with respect to ( p ). So, ( frac{dx}{dp} = a + 2b p ). That means ( dx = (a + 2b p) dp ). Now, since ( p = frac{dy}{dx} ), I can write ( dy = p dx ). Substituting the expression for ( dx ), I get ( dy = p (a + 2b p) dp ). Let me expand that: ( dy = a p dp + 2b p^2 dp ). To find ( y ), I need to integrate both sides with respect to ( p ). So, ( y = int (a p + 2b p^2) dp ). Integrating term by term, the integral of ( a p ) is ( frac{a}{2} p^2 ) and the integral of ( 2b p^2 ) is ( frac{2b}{3} p^3 ). So, putting it all together, ( y = frac{a}{2} p^2 + frac{2b}{3} p^3 + C ), where ( C ) is the constant of integration.But I still have ( p ) in terms of ( x ). From earlier, I have ( x = a p + b p^2 ). So, if I can express ( p ) in terms of ( x ), I can substitute it back into the equation for ( y ). Let me rearrange the equation: ( b p^2 + a p - x = 0 ). This is a quadratic equation in ( p ). Using the quadratic formula, ( p = frac{-a pm sqrt{a^2 + 4b x}}{2b} ).Hmm, so ( p ) can take two values depending on the sign. I guess both solutions are valid unless there's a restriction on ( p ). So, I'll keep both possibilities. Now, substituting this back into the expression for ( y ), I get:( y = frac{a}{2} left( frac{-a pm sqrt{a^2 + 4b x}}{2b} right)^2 + frac{2b}{3} left( frac{-a pm sqrt{a^2 + 4b x}}{2b} right)^3 + C ).This looks a bit complicated. Maybe I can simplify it. Let me compute each term separately. First, the square term:( left( frac{-a pm sqrt{a^2 + 4b x}}{2b} right)^2 = frac{a^2 - 2a sqrt{a^2 + 4b x} + (a^2 + 4b x)}{4b^2} = frac{2a^2 + 4b x - 2a sqrt{a^2 + 4b x}}{4b^2} = frac{a^2 + 2b x - a sqrt{a^2 + 4b x}}{2b^2} ).Now, multiplying by ( frac{a}{2} ):( frac{a}{2} times frac{a^2 + 2b x - a sqrt{a^2 + 4b x}}{2b^2} = frac{a(a^2 + 2b x - a sqrt{a^2 + 4b x})}{4b^2} ).Next, the cube term:( left( frac{-a pm sqrt{a^2 + 4b x}}{2b} right)^3 = frac{(-a)^3 + 3(-a)^2 sqrt{a^2 + 4b x} + 3(-a)(a^2 + 4b x) + (a^2 + 4b x)^{3/2}}{8b^3} ).This seems really messy. Maybe there's a better way to approach this. Perhaps instead of substituting back, I can express ( y ) in terms of ( p ) and then relate it to ( x ). Alternatively, maybe I can parametrize the solution in terms of ( p ) as a parameter.Wait, another thought: since I have ( x ) and ( y ) both expressed in terms of ( p ), maybe I can write the solution parametrically. So, the solution would be:( x = a p + b p^2 ),( y = frac{a}{2} p^2 + frac{2b}{3} p^3 + C ).This way, ( p ) acts as a parameter, and ( x ) and ( y ) are expressed in terms of ( p ). This might be a more straightforward way to present the solution without having to solve for ( p ) explicitly in terms of ( x ).But if I need to express ( y ) solely in terms of ( x ), I might have to deal with the square roots and cube terms, which could complicate things. Maybe it's acceptable to leave the solution in parametric form, especially since solving for ( p ) leads to a complicated expression.Alternatively, perhaps I can consider this as an implicit solution. Let me see. If I have ( x = a p + b p^2 ) and ( y = frac{a}{2} p^2 + frac{2b}{3} p^3 + C ), I can think of ( p ) as a function of ( x ), and ( y ) as a function of ( p ). So, combining these, ( y ) is ultimately a function of ( x ), but it's expressed through the parameter ( p ).I think parametric solutions are acceptable in differential equations, especially when solving explicitly for ( y ) in terms of ( x ) leads to complicated expressions. So, maybe I can present the solution in parametric form as I did above.But just to make sure, let me check if there's another approach. Maybe I can write the original equation as ( b left( frac{dy}{dx} right)^2 + a frac{dy}{dx} - x = 0 ) and treat it as a quadratic in ( frac{dy}{dx} ). Then, solving for ( frac{dy}{dx} ), I get:( frac{dy}{dx} = frac{-a pm sqrt{a^2 + 4b x}}{2b} ).This is consistent with what I found earlier. So, integrating this would give me ( y ) in terms of ( x ), but it's going to involve integrating ( sqrt{a^2 + 4b x} ), which might not be straightforward.Let me try integrating ( frac{dy}{dx} = frac{-a pm sqrt{a^2 + 4b x}}{2b} ). So,( y = int frac{-a pm sqrt{a^2 + 4b x}}{2b} dx ).Let me make a substitution to integrate the square root term. Let ( u = a^2 + 4b x ), then ( du = 4b dx ), so ( dx = frac{du}{4b} ). Then, the integral becomes:( y = frac{-a}{2b} int dx pm frac{1}{2b} int sqrt{u} cdot frac{du}{4b} ).Simplifying,( y = frac{-a}{2b} x pm frac{1}{8b^2} int u^{1/2} du ).Integrating ( u^{1/2} ) gives ( frac{2}{3} u^{3/2} ), so:( y = frac{-a}{2b} x pm frac{1}{8b^2} cdot frac{2}{3} (a^2 + 4b x)^{3/2} + C ).Simplifying further,( y = frac{-a}{2b} x pm frac{1}{12b^2} (a^2 + 4b x)^{3/2} + C ).Hmm, that seems like a valid expression for ( y ) in terms of ( x ). But it's still quite complicated, involving a square root and a cube root. I wonder if this can be simplified more or if there's a better way to present it.Alternatively, going back to the parametric solution, I can think of ( p ) as a parameter and express both ( x ) and ( y ) in terms of ( p ). This might be more elegant and avoid the messy expressions involving square roots.So, summarizing, the solution can be expressed parametrically as:( x = a p + b p^2 ),( y = frac{a}{2} p^2 + frac{2b}{3} p^3 + C ).This seems like a clean and concise way to present the solution without getting bogged down in complicated algebra.But just to make sure, let me verify if differentiating ( y ) with respect to ( x ) gives back the original equation. So, from the parametric equations, ( frac{dy}{dx} = frac{dy/dp}{dx/dp} = frac{a p + 2b p^2}{a + 2b p} ). Simplifying, ( frac{dy}{dx} = p ), which matches our substitution. Then, substituting back into the original equation:( a p + b p^2 = x ), which is exactly the first parametric equation. So, it checks out.Therefore, I think the parametric solution is valid and perhaps the most straightforward way to express the solution to this differential equation.