answer：Okay, so I've got this problem about genetic distribution in beetles, right? It's about the Hardy-Weinberg principle, which I remember is a model that predicts how allele frequencies in a population will remain constant from generation to generation in the absence of other evolutionary influences. The problem has two parts, and I need to tackle them step by step.Starting with part 1: It says that the frequency of the homozygous dominant genotype AA is p squared, and the frequency of the homozygous recessive genotype aa is q squared. I need to write a polynomial equation for the total frequency of the homozygous genotypes and simplify it.Hmm, okay. So, homozygous genotypes are AA and aa. Their frequencies are p² and q² respectively. So, the total frequency should just be the sum of these two, right? So, total homozygous frequency = p² + q². That seems straightforward. But the question says to write a polynomial equation and simplify it. Well, since p and q are allele frequencies and they must add up to 1, I know that q = 1 - p. Maybe I can substitute that into the equation to simplify it further.So, substituting q with (1 - p), the equation becomes p² + (1 - p)². Let me expand that. (1 - p) squared is 1 - 2p + p². So, adding p² to that, we have p² + 1 - 2p + p². Combining like terms, that's 2p² - 2p + 1. So, the polynomial equation is 2p² - 2p + 1. Is that correct? Let me double-check. Yes, p² + q² becomes p² + (1 - p)², which expands to 1 - 2p + 2p². So, yes, 2p² - 2p + 1. That seems right.Moving on to part 2: Dr. Wright observes that the frequency of the heterozygous genotype Aa, which is 2pq, is equal to the sum of the frequencies of both homozygous genotypes, which is p² + q². So, I need to set up an equation where 2pq = p² + q² and then derive a quadratic equation in terms of p and solve for p.Alright, so starting with 2pq = p² + q². Again, since p + q = 1, I can substitute q with (1 - p). Let's do that. So, replacing q with (1 - p), the equation becomes 2p(1 - p) = p² + (1 - p)².Let me expand both sides. On the left side: 2p(1 - p) is 2p - 2p². On the right side: p² + (1 - p)². We already expanded (1 - p)² earlier as 1 - 2p + p². So, adding p², the right side becomes p² + 1 - 2p + p², which simplifies to 2p² - 2p + 1.So now, the equation is 2p - 2p² = 2p² - 2p + 1. Let me bring all terms to one side to form a quadratic equation. Subtracting the left side from both sides: 0 = 2p² - 2p + 1 - (2p - 2p²). Let's distribute the negative sign: 0 = 2p² - 2p + 1 - 2p + 2p². Combining like terms: 2p² + 2p² is 4p², -2p - 2p is -4p, and +1 remains. So, the equation becomes 4p² - 4p + 1 = 0.Wait, is that correct? Let me check my steps again. Starting from 2p - 2p² = 2p² - 2p + 1. If I subtract 2p - 2p² from both sides, it's 0 = 2p² - 2p + 1 - 2p + 2p². Yes, that gives 4p² - 4p + 1 = 0. Hmm, okay.So, now I have a quadratic equation: 4p² - 4p + 1 = 0. To solve for p, I can use the quadratic formula. The quadratic formula is p = [ -b ± sqrt(b² - 4ac) ] / (2a). Here, a = 4, b = -4, c = 1. Plugging these values in: p = [ -(-4) ± sqrt( (-4)² - 4*4*1 ) ] / (2*4). Simplifying: p = [4 ± sqrt(16 - 16)] / 8. The discriminant is sqrt(0), which is 0. So, p = [4 ± 0]/8, which is 4/8, simplifying to 1/2.Wait, so p is 1/2? That seems interesting. So, the only solution is p = 1/2. Let me think about that. If p is 1/2, then q is also 1/2. So, in this case, the allele frequencies are equal. Let me verify if this satisfies the original condition.If p = q = 1/2, then the frequency of AA is (1/2)² = 1/4, the frequency of aa is also (1/2)² = 1/4, and the frequency of Aa is 2*(1/2)*(1/2) = 1/2. So, the sum of homozygous genotypes is 1/4 + 1/4 = 1/2, which is equal to the frequency of heterozygous genotype, which is also 1/2. So, yes, that works. So, p = 1/2 is the solution.But wait, the quadratic equation only gave me one solution, p = 1/2. Usually, quadratic equations have two solutions unless the discriminant is zero, which gives a repeated root. In this case, the discriminant was zero, so we only have one unique solution. So, that makes sense.So, to recap: For part 1, the total frequency of homozygous genotypes is 2p² - 2p + 1. For part 2, solving the equation 2pq = p² + q² leads us to p = 1/2.I think that's it. Let me just make sure I didn't make any arithmetic mistakes. Starting from 2pq = p² + q², substituting q = 1 - p, expanding both sides, moving everything to one side, and solving the quadratic. Yes, it all checks out. So, p must be 1/2.**Final Answer**1. The total frequency of homozygous genotypes is boxed{2p^2 - 2p + 1}.2. The possible allele frequency ( p ) is boxed{dfrac{1}{2}}.

answer：Alright, so I'm trying to figure out this problem where I need to model the relationship between the number of cases solved (CS) and the resources allocated (R) using a quadratic equation. The equation is given as CS = aR² + bR + c, and I have three data points to work with. Let me break this down step by step.First, I need to set up a system of equations using the given data points. The data points are:1. When R = 10, CS = 1502. When R = 20, CS = 3003. When R = 30, CS = 450So, for each of these points, I can plug the values of R and CS into the quadratic equation to get three equations with three unknowns: a, b, and c.Starting with the first data point: R = 10, CS = 150.Plugging into the equation: 150 = a*(10)² + b*(10) + cCalculating 10 squared is 100, so that becomes:150 = 100a + 10b + c ...(Equation 1)Next, the second data point: R = 20, CS = 300.Plugging into the equation: 300 = a*(20)² + b*(20) + c20 squared is 400, so:300 = 400a + 20b + c ...(Equation 2)Third data point: R = 30, CS = 450.Plugging into the equation: 450 = a*(30)² + b*(30) + c30 squared is 900, so:450 = 900a + 30b + c ...(Equation 3)Now, I have three equations:1. 100a + 10b + c = 1502. 400a + 20b + c = 3003. 900a + 30b + c = 450My goal is to solve for a, b, and c. To do this, I can use the method of elimination. Let me subtract Equation 1 from Equation 2 to eliminate c.Equation 2 - Equation 1:(400a - 100a) + (20b - 10b) + (c - c) = 300 - 150That simplifies to:300a + 10b = 150 ...(Equation 4)Similarly, subtract Equation 2 from Equation 3:(900a - 400a) + (30b - 20b) + (c - c) = 450 - 300Which simplifies to:500a + 10b = 150 ...(Equation 5)Now, I have two equations:4. 300a + 10b = 1505. 500a + 10b = 150Hmm, interesting. Both Equation 4 and Equation 5 have the same right-hand side, 150, but different coefficients for a. Let me subtract Equation 4 from Equation 5 to eliminate b.Equation 5 - Equation 4:(500a - 300a) + (10b - 10b) = 150 - 150Simplifying:200a = 0So, 200a = 0 implies that a = 0.Wait, if a is 0, then the quadratic equation reduces to a linear equation, which is CS = bR + c. That's interesting. Let me check if this makes sense with the data points.Looking back at the data points:When R increases by 10 (from 10 to 20), CS increases by 150 (from 150 to 300). Similarly, when R increases by another 10 (from 20 to 30), CS increases by another 150 (from 300 to 450). So, the increase in CS is linear with respect to R. That suggests that the relationship is indeed linear, not quadratic. So, a being 0 makes sense.Now, knowing that a = 0, I can substitute back into Equation 4 or Equation 5 to find b.Let's use Equation 4: 300a + 10b = 150Substituting a = 0:300*0 + 10b = 150So, 10b = 150Dividing both sides by 10:b = 15Now, with a = 0 and b = 15, I can substitute into Equation 1 to find c.Equation 1: 100a + 10b + c = 150Substituting a = 0 and b = 15:100*0 + 10*15 + c = 150Which is:0 + 150 + c = 150So, 150 + c = 150Subtracting 150 from both sides:c = 0So, the quadratic equation simplifies to CS = 0*R² + 15*R + 0, which is just CS = 15R.Let me verify this with the given data points to ensure it's correct.For R = 10: CS = 15*10 = 150 ✔️For R = 20: CS = 15*20 = 300 ✔️For R = 30: CS = 15*30 = 450 ✔️Perfect, it all checks out. So, the constants are a = 0, b = 15, and c = 0.Now, moving on to part 2. The detective claims that by increasing resources to 40, the number of cases solved will exceed 700. I need to verify this using the quadratic model.But wait, our quadratic model turned out to be linear because a = 0. So, the model is CS = 15R.Let me compute CS when R = 40.CS = 15*40 = 600.So, according to the model, when R = 40, CS = 600.But the detective claims that CS will exceed 700. However, according to the model, it's only 600, which is less than 700.Therefore, the detective's claim is not supported by the model. The model predicts that at R = 40, CS = 600, which is below 700.But just to be thorough, let me also check if I made any mistakes in my calculations. Let me go through the steps again.Starting with the three equations:1. 100a + 10b + c = 1502. 400a + 20b + c = 3003. 900a + 30b + c = 450Subtracting Equation 1 from Equation 2:300a + 10b = 150Subtracting Equation 2 from Equation 3:500a + 10b = 150Subtracting these two results:200a = 0 => a = 0Then, substituting a = 0 into 300a + 10b = 150:10b = 150 => b = 15Then, substituting a = 0 and b = 15 into Equation 1:100*0 + 10*15 + c = 150 => 150 + c = 150 => c = 0So, the model is indeed CS = 15R.At R = 40, CS = 15*40 = 600.Therefore, the detective's claim that CS will exceed 700 when R = 40 is incorrect according to the model.Alternatively, if I consider the possibility that the relationship might not be perfectly linear, but given the data points, the model is linear. So, unless there's a change in the relationship beyond the given data, the model doesn't support the detective's claim.I think that's all. I don't see any mistakes in my calculations, so I'm confident in the results.

answer：First, I need to determine the minimum amount of RAM the user should have available to install and run the software smoothly. The software requires a minimum of 8 GB of RAM, and the installation process also needs an additional 2 GB for other processes. Therefore, the total required RAM is 8 + 2 = 10 GB. This means the user's available RAM, x, must be at least 10 GB. So, the inequality is x ≥ 10.Next, I'll solve the polynomial equation ( x^2 - 4x + 4 = 0 ). To do this, I can factor the equation. It looks like a perfect square trinomial, so it factors to ( (x - 2)^2 = 0 ). Solving for x gives x = 2. Now, I need to check if this value satisfies the inequality from the first part. The solution x = 2 is much less than the required 10 GB, so it does not satisfy the inequality. This means the user's system does not have enough RAM to successfully install and run the software smoothly.

answer：Okay, so I have this problem about skincare products and their effectiveness based on concentration. Let me try to figure it out step by step.First, the woman is looking at a quadratic equation for effectiveness, which is E(c) = -2c² + 8c. She wants to know the concentration where the effectiveness is maximized. Hmm, quadratic equations have either a maximum or minimum value, depending on the coefficient of the c² term. Since the coefficient here is -2, which is negative, the parabola opens downward, meaning it has a maximum point. That makes sense because too high a concentration might not be good, so effectiveness peaks somewhere in the middle.To find the maximum, I remember that for a quadratic equation in the form E(c) = ax² + bx + c, the vertex occurs at c = -b/(2a). So in this case, a is -2 and b is 8. Plugging those in, c = -8/(2*(-2)) = -8/(-4) = 2. So, the concentration that maximizes effectiveness is 2%. That seems straightforward.Now, moving on to the second part. She's comparing two products, A and B. Product A's effectiveness is the same as before, E_A(c) = -2c² + 8c, and Product B is E_B(c) = -c² + 6c + 9. She wants to know for what concentration values Product A is more effective than Product B. So, I need to solve the inequality E_A(c) > E_B(c).Let me write that out: -2c² + 8c > -c² + 6c + 9. To solve this, I should bring all terms to one side so I can have a standard quadratic inequality. Subtracting E_B(c) from both sides, I get:-2c² + 8c - (-c² + 6c + 9) > 0Simplifying that, distribute the negative sign:-2c² + 8c + c² - 6c - 9 > 0Combine like terms:(-2c² + c²) + (8c - 6c) - 9 > 0Which simplifies to:- c² + 2c - 9 > 0Hmm, so the inequality is -c² + 2c - 9 > 0. It might be easier to work with if I multiply both sides by -1 to make the coefficient of c² positive. But I have to remember that multiplying both sides of an inequality by a negative number reverses the inequality sign. So:c² - 2c + 9 < 0Now, I have c² - 2c + 9 < 0. Let me analyze this quadratic. The quadratic is c² - 2c + 9. To find where it's less than zero, I need to find its roots, if any. Let's compute the discriminant: D = b² - 4ac = (-2)² - 4*1*9 = 4 - 36 = -32.Since the discriminant is negative, the quadratic doesn't cross the x-axis and doesn't have real roots. Since the coefficient of c² is positive, the parabola opens upward. So, the quadratic is always positive because it doesn't cross the x-axis and opens upward. Therefore, c² - 2c + 9 is always greater than zero for all real numbers c. That means the inequality c² - 2c + 9 < 0 has no solution.Wait, so going back, that means the original inequality -c² + 2c - 9 > 0 also has no solution because we saw that it's equivalent to c² - 2c + 9 < 0, which is never true. Therefore, Product A is never more effective than Product B. Is that possible?Let me double-check my steps. Starting with E_A(c) > E_B(c):-2c² + 8c > -c² + 6c + 9Subtracting E_B(c):-2c² + 8c + c² - 6c - 9 > 0Simplify:- c² + 2c - 9 > 0Multiply by -1 (and reverse inequality):c² - 2c + 9 < 0Discriminant: (-2)^2 - 4*1*9 = 4 - 36 = -32 < 0So, no real roots, and since it's positive definite, the inequality c² - 2c + 9 < 0 is never true. Therefore, E_A(c) is never greater than E_B(c). So, Product A is never more effective than Product B. That seems a bit counterintuitive, but mathematically, it checks out.But just to be thorough, let me plug in some values to see. Let's take c = 0:E_A(0) = 0, E_B(0) = 9. So, 0 < 9. Correct.c = 1:E_A(1) = -2 + 8 = 6E_B(1) = -1 + 6 + 9 = 146 < 14. Correct.c = 2:E_A(2) = -8 + 16 = 8E_B(2) = -4 + 12 + 9 = 178 < 17. Correct.c = 3:E_A(3) = -18 + 24 = 6E_B(3) = -9 + 18 + 9 = 186 < 18. Correct.c = 4:E_A(4) = -32 + 32 = 0E_B(4) = -16 + 24 + 9 = 170 < 17. Correct.c = 5:E_A(5) = -50 + 40 = -10E_B(5) = -25 + 30 + 9 = 14-10 < 14. Correct.So, in all these test cases, E_A(c) is less than E_B(c). Therefore, it seems that Product B is always more effective than Product A across all concentrations. So, the answer to the second question is that there are no concentration values where Product A is more effective than Product B.Wait, but the original inequality was E_A(c) > E_B(c). If there are no solutions, then the set of c where this holds is empty. So, the answer is that there are no such concentrations.But just to make sure I didn't make a mistake in the algebra:Starting with E_A(c) > E_B(c):-2c² + 8c > -c² + 6c + 9Bring all terms to left:-2c² + 8c + c² - 6c - 9 > 0Simplify:(-2c² + c²) + (8c - 6c) - 9 > 0Which is:- c² + 2c - 9 > 0Multiply both sides by -1 (reverse inequality):c² - 2c + 9 < 0Quadratic equation c² - 2c + 9 = 0 has discriminant D = 4 - 36 = -32 < 0, so no real roots. Since the coefficient of c² is positive, the quadratic is always positive. Therefore, c² - 2c + 9 is always positive, so the inequality c² - 2c + 9 < 0 is never true. Therefore, there are no solutions. So, Product A is never more effective than Product B.Yeah, that seems correct.**Final Answer**1. The concentration that maximizes effectiveness is boxed{2}%.2. There are no concentration values where Product A is more effective than Product B, so the solution is boxed{text{None}}.

answer：First, I'll define the variables based on the information provided. Let ( x ) represent the number of science courses.Next, I'll express the number of math and literature courses in terms of ( x ). Since there are 3 more math courses than science courses, the number of math courses will be ( x + 3 ). Additionally, there are twice as many literature courses as science courses, so the number of literature courses will be ( 2x ).Now, I'll set up the equation for the total number of courses. The sum of math, science, and literature courses should equal 18:[(x + 3) + x + 2x = 18]Combining like terms gives:[4x + 3 = 18]Subtracting 3 from both sides:[4x = 15]Finally, dividing both sides by 4 to solve for ( x ):[x = frac{15}{4}]This means there are ( frac{15}{4} ) science courses, ( frac{27}{4} ) math courses, and ( frac{15}{2} ) literature courses.

answer：Alright, so I have this problem about setting up a swimming relay race. Let me try to figure it out step by step. First, the pool is 50 meters long, and each team needs to complete exactly 200 meters. Each team has 4 swimmers, and each swimmer does an equal distance. I need to find out how many laps each swimmer must complete. Hmm, okay.Let me think about what a lap is. In swimming, a lap is typically one length of the pool, right? So, if the pool is 50 meters, then one lap is 50 meters. But wait, sometimes people consider a lap as going and coming back, which would be 100 meters. Hmm, the problem doesn't specify, so I need to clarify that.Wait, the problem says "each swimmer completes an equal distance." It doesn't specify whether a lap is one length or two. Hmm. Maybe I should assume it's one length, so 50 meters per lap. That seems more straightforward. So, each swimmer swims x laps, each lap being 50 meters. So, each swimmer swims 50x meters. Since there are 4 swimmers, the total distance for the team is 4 times that. So, 4 * 50x = 200x meters. But the team needs to complete exactly 200 meters. So, 200x = 200. Wait, that seems too easy. Solving for x, we get x = 1. So, each swimmer swims 1 lap. That seems correct. But let me double-check.If each swimmer swims 1 lap (50 meters), then four swimmers would do 4 * 50 = 200 meters. Yep, that adds up. So, each swimmer must complete 1 lap. But wait, another thought: sometimes in relays, each swimmer does multiple laps. Maybe I misinterpreted the lap. If a lap is considered as two lengths (going and coming back), then each lap would be 100 meters. Let me see if that changes the answer.If a lap is 100 meters, then each swimmer swims x laps, so 100x meters. Four swimmers would do 4 * 100x = 400x meters. The total needs to be 200 meters, so 400x = 200. Solving for x, we get x = 0.5. Hmm, half a lap. That seems a bit odd because you can't really have half a lap in a relay race. Swimmers usually do whole laps. So, maybe my initial assumption was correct that a lap is 50 meters.Alternatively, maybe the problem defines a lap as one length, so 50 meters. That would make each swimmer do 1 lap, which is 50 meters, and four swimmers make 200 meters. That seems more practical.So, I think the first part is solved: each swimmer must complete 1 lap. So, x = 1. The equation would be 4 * 50x = 200, which simplifies to 200x = 200, so x = 1.Moving on to the second problem. The total number of lanes available is represented by n, and it's given by the quadratic equation n² - 10n + 24 = 0. I need to solve this quadratic equation to find the possible number of lanes.Okay, quadratic equations can be solved by factoring, completing the square, or using the quadratic formula. Let me try factoring first because it's usually quicker if it factors nicely.Looking at n² - 10n + 24, I need two numbers that multiply to 24 and add up to -10. Let's see: factors of 24 are 1 & 24, 2 & 12, 3 & 8, 4 & 6. Hmm, 4 and 6 multiply to 24, and if both are negative, they add up to -10. So, yes, -4 and -6.So, the equation factors as (n - 4)(n - 6) = 0. Therefore, the solutions are n = 4 and n = 6. Let me double-check by expanding (n - 4)(n - 6). That gives n² - 6n - 4n + 24 = n² - 10n + 24. Yep, that's correct.So, the possible number of lanes available is either 4 or 6.But wait, in a swimming pool, the number of lanes is typically an even number because each lane is on either side of the pool. So, 4 or 6 lanes are both possible. 4 lanes would mean two on each side, and 6 lanes would mean three on each side. So, both are feasible.Therefore, the solutions are n = 4 and n = 6.So, summarizing:1. Each swimmer must complete 1 lap.2. The possible number of lanes available is 4 or 6.**Final Answer**1. The number of laps each swimmer must complete is boxed{1}.2. The possible number of lanes available is boxed{4} and boxed{6}.