💡Okay, so I need to calculate the product of (3 + i) and i. Hmm, let's see. I remember that when you multiply complex numbers, you distribute the multiplication, kind of like the FOIL method for binomials. So, I think I should multiply each term in the first parentheses by each term in the second parentheses.Alright, let's write it out step by step. The expression is (3 + i) multiplied by i. So, I can think of it as 3 times i plus i times i. That would be 3i plus i squared. Wait, what's i squared? Oh, right, i squared is equal to -1 because i is the imaginary unit defined by the property that i squared equals -1.So, substituting that back in, I have 3i plus (-1). That simplifies to -1 plus 3i. Let me double-check that to make sure I didn't make a mistake. Multiplying (3 + i) by i: 3 times i is 3i, and i times i is i squared, which is -1. So, adding those together gives -1 + 3i. Yeah, that seems right.I guess another way to think about it is to treat it like a distributive property problem. Multiply i by 3, which is 3i, and then multiply i by i, which is i squared, and then combine like terms. Since i squared is -1, it becomes -1 + 3i. Yeah, that makes sense.Wait, could there be another method to verify this? Maybe using the rectangular form of complex numbers. If I represent (3 + i) as a complex number in the form a + bi, where a is 3 and b is 1, and then multiply it by i, which is 0 + 1i. There's a formula for multiplying two complex numbers: (a + bi)(c + di) = (ac - bd) + (ad + bc)i.Applying that formula here, a is 3, b is 1, c is 0, and d is 1. So, the real part would be (3 * 0 - 1 * 1) which is 0 - 1 = -1. The imaginary part would be (3 * 1 + 1 * 0) which is 3 + 0 = 3. So, combining those, we get -1 + 3i, which matches what I got earlier. That's reassuring.I think I'm confident now that the product of (3 + i) and i is indeed -1 + 3i. It's always good to check using different methods to make sure the answer is consistent. This helps prevent any mistakes that might come from misapplying the multiplication rules for complex numbers.Another thing I could do is plot these complex numbers on the complex plane to visualize the multiplication. The complex number (3 + i) is located at the point (3, 1) in the plane, and multiplying by i is equivalent to rotating the point 90 degrees counterclockwise around the origin. So, rotating (3, 1) by 90 degrees should give me a new point.To rotate a point (a, b) 90 degrees counterclockwise, the new coordinates become (-b, a). Applying that to (3, 1), the new point would be (-1, 3), which corresponds to the complex number -1 + 3i. That's another way to see that the result is correct. It's interesting how different approaches can lead to the same answer.I wonder if there's a geometric interpretation of multiplying by i. From what I understand, multiplying by i rotates a complex number by 90 degrees in the complex plane. So, in this case, multiplying (3 + i) by i rotates it 90 degrees, resulting in -1 + 3i. That makes sense because the real part becomes the negative of the original imaginary part, and the imaginary part becomes the original real part.This rotation idea is pretty cool. It helps me visualize what's happening when I multiply complex numbers. Instead of just crunching numbers, I can think about how the points move in the plane. This might be useful for understanding more complex operations with complex numbers in the future.Let me summarize the steps I took to solve this problem:1. Recognize that multiplying (3 + i) by i requires distributing the multiplication.2. Apply the distributive property: (3 + i)i = 3i + i^2.3. Recall that i^2 = -1, so substitute that in: 3i + (-1) = -1 + 3i.4. Verify the result using the formula for multiplying complex numbers: (a + bi)(c + di) = (ac - bd) + (ad + bc)i.5. Confirm the answer by considering the geometric interpretation of multiplication by i as a rotation in the complex plane.By going through these steps, I ensured that my answer was correct and reinforced my understanding of complex number multiplication. It's helpful to approach problems from multiple angles, especially when dealing with concepts that have both algebraic and geometric interpretations.I also think it's important to remember the fundamental properties of complex numbers, like the definition of i, because they form the basis for more advanced operations. Without a solid grasp of these basics, it can be challenging to tackle more complex problems down the line.In conclusion, calculating (3 + i)i involves distributing the multiplication, applying the definition of i squared, and simplifying the result. By verifying the answer through different methods, I can be confident in its correctness. This exercise also highlighted the value of understanding both algebraic and geometric perspectives when working with complex numbers.

💡To solve the problem, we need to calculate the answers for both Amy and Ben and then find the difference between their results.1. **Calculate Amy’s Answer (A):** - Amy follows the correct order of operations (PEMDAS/BODMAS), which means she performs the multiplication inside the parentheses first. [ A = 12 - (3 times 4) = 12 - 12 = 0 ]2. **Calculate Ben’s Answer (B):** - Ben ignores the parentheses and instead subtracts 3 from 12 first and then multiplies the result by 4. [ B = (12 - 3) times 4 = 9 times 4 = 36 ]3. **Find the Difference (A - B):** - Subtract Ben’s result from Amy’s result. [ A - B = 0 - 36 = -36 ]The difference between Amy’s and Ben’s results is (-36).

💡First, I need to determine the relationship between the length and the width of the rectangular field. The problem states that the length is double the width, so I can express the length as ( l = 2w ).Next, I'll calculate the area of the rectangular field using the formula for the area of a rectangle, which is length multiplied by width:[text{Area of field} = l times w]The problem also mentions a square-shaped pond with a side length of 6 meters. The area of the pond can be calculated as:[text{Area of pond} = 6 times 6 = 36 text{ m}^2]According to the problem, the area of the pond is ( frac{1}{8} ) of the area of the field. This gives me the equation:[36 = frac{1}{8} times (l times w)]Substituting ( l = 2w ) into the equation:[36 = frac{1}{8} times (2w times w)][36 = frac{2w^2}{8}][36 = frac{w^2}{4}][144 = w^2]Taking the square root of both sides to find the width:[w = sqrt{144} = 12 text{ m}]Finally, I'll find the length by doubling the width:[l = 2w = 2 times 12 = 24 text{ m}]So, the length of the field is 24 meters.

💡Alright, so I have this equation here: [ a(a - b) + b(b - c) + c(c - a) = 0 ]And I need to figure out under which of the given conditions the integers ( a ), ( b ), and ( c ) satisfy this equation. The options are labeled from A to E, each providing specific relationships between ( a ), ( b ), and ( c ).First, I want to understand the equation better. Let me try expanding it step by step to see if I can simplify it or find a pattern.Starting with the original equation:[ a(a - b) + b(b - c) + c(c - a) = 0 ]Let me distribute each term:- ( a(a - b) = a^2 - ab )- ( b(b - c) = b^2 - bc )- ( c(c - a) = c^2 - ca )So, substituting these back into the equation:[ a^2 - ab + b^2 - bc + c^2 - ca = 0 ]Now, let me rearrange the terms to group similar ones together:[ a^2 + b^2 + c^2 - ab - bc - ca = 0 ]Hmm, this looks familiar. I think this expression can be rewritten in terms of squared differences. Let me recall that:[ (a - b)^2 + (b - c)^2 + (c - a)^2 ]Expanding each square:- ( (a - b)^2 = a^2 - 2ab + b^2 )- ( (b - c)^2 = b^2 - 2bc + c^2 )- ( (c - a)^2 = c^2 - 2ca + a^2 )Adding these together:[ (a^2 - 2ab + b^2) + (b^2 - 2bc + c^2) + (c^2 - 2ca + a^2) ][ = 2a^2 + 2b^2 + 2c^2 - 2ab - 2bc - 2ca ]So, if I factor out a 2:[ 2(a^2 + b^2 + c^2 - ab - bc - ca) ]Comparing this with my earlier rearranged equation:[ a^2 + b^2 + c^2 - ab - bc - ca = 0 ]It seems that:[ (a - b)^2 + (b - c)^2 + (c - a)^2 = 2 times 0 = 0 ]Wait, that can't be right. Let me double-check. If I have:[ (a - b)^2 + (b - c)^2 + (c - a)^2 = 2(a^2 + b^2 + c^2 - ab - bc - ca) ]So, if ( a^2 + b^2 + c^2 - ab - bc - ca = 0 ), then:[ (a - b)^2 + (b - c)^2 + (c - a)^2 = 0 ]Ah, that makes sense. Since each squared term is non-negative, the only way their sum can be zero is if each individual squared term is zero. Therefore:[ (a - b)^2 = 0 ][ (b - c)^2 = 0 ][ (c - a)^2 = 0 ]Which implies:[ a = b ][ b = c ][ c = a ]So, all three variables must be equal: ( a = b = c ).Now, I need to check each of the given conditions to see under which one ( a = b = c ) holds.Let's go through each option one by one.**Option A: ( a > b ) and ( b = c + 1 )**If ( b = c + 1 ), then ( c = b - 1 ). Also, ( a > b ) implies ( a ) is at least ( b + 1 ). So, ( a ) is greater than ( b ), which is greater than ( c ). Therefore, ( a ), ( b ), and ( c ) cannot all be equal. This condition does not satisfy the equation.**Option B: ( a = b + 1 ) and ( b = c + 2 )**From ( b = c + 2 ), we get ( c = b - 2 ). Then, ( a = b + 1 ). So, ( a ) is greater than ( b ), and ( b ) is greater than ( c ). Again, ( a ), ( b ), and ( c ) are distinct, so they cannot all be equal. This condition does not satisfy the equation.**Option C: ( a = c + 1 ) and ( b = a + 2 )**From ( a = c + 1 ), we have ( c = a - 1 ). Then, ( b = a + 2 ). So, ( b ) is greater than ( a ), which is greater than ( c ). Thus, ( a ), ( b ), and ( c ) are distinct, so they cannot all be equal. This condition does not satisfy the equation.**Option D: ( a = c ) and ( b - 2 = c )**From ( a = c ), we have ( a = c ). From ( b - 2 = c ), we get ( b = c + 2 ). So, ( b = a + 2 ). Therefore, ( a = c ), ( b = a + 2 ). The only way all three can be equal is if ( a = b = c ). Let's see if that's possible.If ( a = b = c ), then substituting into ( b = a + 2 ) gives:[ a = a + 2 ][ 0 = 2 ]Which is a contradiction. Wait, that doesn't make sense. Did I make a mistake?Hold on, if ( a = c ) and ( b = c + 2 ), then ( b = a + 2 ). So, ( b ) is always 2 more than ( a ). Therefore, ( a ), ( b ), and ( c ) cannot all be equal unless ( a = b = c ), which would require ( a = a + 2 ), which is impossible unless we're working modulo something, but since we're dealing with integers, this can't happen.Wait, maybe I misinterpreted the condition. Let me read it again: "a = c and b - 2 = c". So, ( a = c ) and ( b = c + 2 ). So, ( a = c ), ( b = a + 2 ). Therefore, ( a ), ( b ), and ( c ) can only be equal if ( a = a + 2 ), which is impossible. So, actually, this condition also doesn't allow ( a = b = c ). Hmm, this is confusing because earlier I thought this might be the answer, but now I'm not sure.Wait, maybe I need to reconsider. The equation requires ( a = b = c ). So, if the condition allows for ( a = b = c ), then it's a possible answer. But in this case, the condition is ( a = c ) and ( b = c + 2 ). So, unless ( c + 2 = c ), which is impossible, ( b ) cannot equal ( c ). Therefore, ( a ), ( b ), and ( c ) cannot all be equal under this condition either. So, this condition doesn't satisfy the equation.Wait, but the original equation requires ( a = b = c ). So, if none of the conditions allow ( a = b = c ), then none of them satisfy the equation. But that can't be right because the problem is asking which condition makes the equation true.Wait, maybe I made a mistake in my initial analysis. Let me go back.I had:[ a(a - b) + b(b - c) + c(c - a) = 0 ]Which I expanded to:[ a^2 - ab + b^2 - bc + c^2 - ca = 0 ]Then, I rearranged it as:[ a^2 + b^2 + c^2 - ab - bc - ca = 0 ]And then I recognized that this is equal to:[ frac{1}{2} left( (a - b)^2 + (b - c)^2 + (c - a)^2 right) = 0 ]Wait, actually, earlier I thought it was equal to the sum of squares, but now I'm recalling that:[ a^2 + b^2 + c^2 - ab - bc - ca = frac{1}{2} left( (a - b)^2 + (b - c)^2 + (c - a)^2 right) ]So, if that's the case, then:[ frac{1}{2} left( (a - b)^2 + (b - c)^2 + (c - a)^2 right) = 0 ]Which still implies that each squared term is zero, so ( a = b = c ).Therefore, my initial conclusion was correct: ( a = b = c ) is necessary for the equation to hold.But then, looking back at the options, none of them seem to allow ( a = b = c ). Wait, let me check again.**Option D: ( a = c ) and ( b - 2 = c )**So, ( a = c ) and ( b = c + 2 ). Therefore, ( b = a + 2 ). So, ( a = c ), ( b = a + 2 ). The only way ( a = b = c ) is if ( a = a + 2 ), which is impossible. So, this condition doesn't allow ( a = b = c ).Wait, but maybe I'm missing something. If ( a = c ) and ( b = c + 2 ), then ( a = c ), ( b = a + 2 ). So, ( a ), ( b ), and ( c ) are in an arithmetic sequence with a common difference of 2. So, ( a = c ), ( b = c + 2 ). Therefore, ( a ), ( b ), ( c ) are distinct unless the common difference is zero, which it's not.So, none of the options seem to allow ( a = b = c ). But that can't be, because the problem is asking which condition makes the equation true. So, maybe my initial analysis is wrong.Wait, let me try plugging in the conditions into the original equation to see if they satisfy it, instead of assuming ( a = b = c ).Let me start with **Option A: ( a > b ) and ( b = c + 1 )**Let me assign some values. Let's say ( c = 1 ), then ( b = 2 ), and ( a ) must be greater than 2, say ( a = 3 ).Plugging into the equation:[ 3(3 - 2) + 2(2 - 1) + 1(1 - 3) ][ = 3(1) + 2(1) + 1(-2) ][ = 3 + 2 - 2 = 3 neq 0 ]So, it doesn't satisfy the equation. Let me try another set. ( c = 0 ), ( b = 1 ), ( a = 2 ).[ 2(2 - 1) + 1(1 - 0) + 0(0 - 2) ][ = 2(1) + 1(1) + 0(-2) ][ = 2 + 1 + 0 = 3 neq 0 ]Still not zero. Maybe another one. ( c = -1 ), ( b = 0 ), ( a = 1 ).[ 1(1 - 0) + 0(0 - (-1)) + (-1)(-1 - 1) ][ = 1(1) + 0(1) + (-1)(-2) ][ = 1 + 0 + 2 = 3 neq 0 ]Hmm, seems like it's consistently giving 3. Maybe this condition always results in 3? Let me check algebraically.Given ( b = c + 1 ) and ( a > b ), let me express everything in terms of ( c ).Let ( b = c + 1 ), ( a = c + k ) where ( k geq 2 ) (since ( a > b = c + 1 )).Plugging into the equation:[ (c + k)( (c + k) - (c + 1) ) + (c + 1)( (c + 1) - c ) + c( c - (c + k) ) ]Simplify each term:First term:[ (c + k)(k - 1) ]Second term:[ (c + 1)(1) = c + 1 ]Third term:[ c(-k) = -ck ]So, combining all terms:[ (c + k)(k - 1) + (c + 1) - ck ]Expand the first term:[ c(k - 1) + k(k - 1) + c + 1 - ck ][ = ck - c + k^2 - k + c + 1 - ck ]Simplify:- ( ck - ck = 0 )- ( -c + c = 0 )- Remaining: ( k^2 - k + 1 )So, the equation simplifies to:[ k^2 - k + 1 = 0 ]But ( k ) is an integer greater than or equal to 2. Let's plug in ( k = 2 ):[ 4 - 2 + 1 = 3 neq 0 ]( k = 3 ):[ 9 - 3 + 1 = 7 neq 0 ]And so on. It's always positive and increasing. So, this condition never satisfies the equation. So, Option A is out.**Option B: ( a = b + 1 ) and ( b = c + 2 )**Express everything in terms of ( c ).Let ( b = c + 2 ), then ( a = (c + 2) + 1 = c + 3 ).So, ( a = c + 3 ), ( b = c + 2 ), ( c = c ).Plug into the equation:[ (c + 3)( (c + 3) - (c + 2) ) + (c + 2)( (c + 2) - c ) + c( c - (c + 3) ) ]Simplify each term:First term:[ (c + 3)(1) = c + 3 ]Second term:[ (c + 2)(2) = 2c + 4 ]Third term:[ c(-3) = -3c ]Combine all terms:[ (c + 3) + (2c + 4) - 3c ][ = c + 3 + 2c + 4 - 3c ][ = (c + 2c - 3c) + (3 + 4) ][ = 0c + 7 = 7 neq 0 ]So, regardless of ( c ), the equation equals 7, which is not zero. Therefore, Option B does not satisfy the equation.**Option C: ( a = c + 1 ) and ( b = a + 2 )**Express everything in terms of ( c ).Let ( a = c + 1 ), then ( b = (c + 1) + 2 = c + 3 ).So, ( a = c + 1 ), ( b = c + 3 ), ( c = c ).Plug into the equation:[ (c + 1)( (c + 1) - (c + 3) ) + (c + 3)( (c + 3) - c ) + c( c - (c + 1) ) ]Simplify each term:First term:[ (c + 1)(-2) = -2c - 2 ]Second term:[ (c + 3)(3) = 3c + 9 ]Third term:[ c(-1) = -c ]Combine all terms:[ (-2c - 2) + (3c + 9) - c ][ = (-2c + 3c - c) + (-2 + 9) ][ = 0c + 7 = 7 neq 0 ]Again, the equation equals 7, which is not zero. So, Option C is out.**Option D: ( a = c ) and ( b - 2 = c )**Express everything in terms of ( c ).Let ( a = c ), and ( b = c + 2 ).So, ( a = c ), ( b = c + 2 ), ( c = c ).Plug into the equation:[ c(c - (c + 2)) + (c + 2)((c + 2) - c) + c(c - c) ]Simplify each term:First term:[ c(-2) = -2c ]Second term:[ (c + 2)(2) = 2c + 4 ]Third term:[ c(0) = 0 ]Combine all terms:[ -2c + 2c + 4 + 0 ][ = 0c + 4 = 4 neq 0 ]Hmm, so the equation equals 4, which is not zero. So, Option D also doesn't satisfy the equation.Wait, but earlier I thought that maybe if ( a = b = c ), it would work, but under Option D, ( a = c ) and ( b = c + 2 ), so ( a ), ( b ), ( c ) can't all be equal. So, that's consistent.**Option E: ( a + b + c = 0 )**This is a different condition. Let me see if this can satisfy the equation.Given ( a + b + c = 0 ), let me express one variable in terms of the others. Let's say ( c = -a - b ).Plug into the equation:[ a(a - b) + b(b - (-a - b)) + (-a - b)((-a - b) - a) ]Simplify each term:First term:[ a(a - b) = a^2 - ab ]Second term:[ b(b + a + b) = b(2b + a) = 2b^2 + ab ]Third term:[ (-a - b)(-a - b - a) = (-a - b)(-2a - b) ]Let me expand this:[ (-a)(-2a - b) + (-b)(-2a - b) ][ = 2a^2 + ab + 2ab + b^2 ][ = 2a^2 + 3ab + b^2 ]Now, combine all terms:First term: ( a^2 - ab )Second term: ( 2b^2 + ab )Third term: ( 2a^2 + 3ab + b^2 )Adding them together:[ (a^2 - ab) + (2b^2 + ab) + (2a^2 + 3ab + b^2) ]Combine like terms:- ( a^2 + 2a^2 = 3a^2 )- ( -ab + ab + 3ab = 3ab )- ( 2b^2 + b^2 = 3b^2 )So, total:[ 3a^2 + 3ab + 3b^2 ]Factor out a 3:[ 3(a^2 + ab + b^2) ]For this to equal zero, we need:[ a^2 + ab + b^2 = 0 ]Since ( a ) and ( b ) are integers, the only solution is ( a = b = 0 ). Therefore, ( c = -a - b = 0 ). So, ( a = b = c = 0 ).Therefore, under condition E, if ( a + b + c = 0 ), the equation is satisfied only when all three are zero. So, this condition can satisfy the equation, but only in the trivial case where all are zero.But wait, the problem is asking under which conditions the integers ( a ), ( b ), and ( c ) satisfy the equation. So, if ( a + b + c = 0 ), then the equation is satisfied only when ( a = b = c = 0 ). So, it's a possible condition, but it's a very specific case.However, looking back at the options, Option D also led to a specific case where the equation didn't hold unless all variables were equal, which they couldn't be under that condition. So, Option E is the only one that can satisfy the equation, albeit only in the trivial case.But wait, earlier when I plugged in Option D, I got 4, which is not zero, so it doesn't satisfy the equation. Option E, on the other hand, can satisfy the equation if ( a = b = c = 0 ). So, Option E is a valid condition that can satisfy the equation.But wait, the problem is asking under which of the following conditions the equation is satisfied. So, if the condition is ( a + b + c = 0 ), then the equation is satisfied only when ( a = b = c = 0 ). So, it's a possible condition, but it's not general.But looking back at the problem statement, it says "integers ( a ), ( b ), and ( c ) satisfy the equation". So, it's not specifying that they have to be non-zero or anything. So, if ( a + b + c = 0 ), then the equation is satisfied when ( a = b = c = 0 ). So, this is a valid condition.But earlier, when I thought ( a = b = c ) was necessary, I concluded that none of the conditions allow ( a = b = c ), but actually, Option E allows ( a = b = c = 0 ), which is a specific case where they are equal.So, in that case, Option E is the correct answer because it allows the equation to hold when ( a = b = c = 0 ).Wait, but earlier when I plugged in Option E with ( a = b = c = 0 ), it worked. Let me confirm:[ 0(0 - 0) + 0(0 - 0) + 0(0 - 0) = 0 ]Yes, that's correct. So, Option E is valid.But wait, in my earlier analysis, I thought that ( a = b = c ) was necessary, but actually, it's sufficient but not necessary? Or is it both necessary and sufficient?Wait, from the equation:[ a^2 + b^2 + c^2 - ab - bc - ca = 0 ]Which is equivalent to:[ frac{1}{2} left( (a - b)^2 + (b - c)^2 + (c - a)^2 right) = 0 ]Which implies ( a = b = c ). So, ( a = b = c ) is both necessary and sufficient for the equation to hold.Therefore, the only way the equation holds is if ( a = b = c ). So, any condition that allows ( a = b = c ) would satisfy the equation.Looking back at the options:- **Option A**: ( a > b ) and ( b = c + 1 ). This implies ( a > b > c ), so ( a neq b neq c ). Doesn't allow ( a = b = c ).- **Option B**: ( a = b + 1 ) and ( b = c + 2 ). So, ( a = c + 3 ), ( b = c + 2 ). Thus, ( a neq b neq c ). Doesn't allow ( a = b = c ).- **Option C**: ( a = c + 1 ) and ( b = a + 2 ). So, ( a = c + 1 ), ( b = c + 3 ). Thus, ( a neq b neq c ). Doesn't allow ( a = b = c ).- **Option D**: ( a = c ) and ( b = c + 2 ). So, ( a = c ), ( b = c + 2 ). Thus, ( a neq b ). Doesn't allow ( a = b = c ).- **Option E**: ( a + b + c = 0 ). This can be satisfied if ( a = b = c = 0 ), which is the only case where ( a = b = c ). So, this condition allows ( a = b = c ) in the trivial case.Therefore, the only condition that allows the equation to hold is Option E, because it's the only one that permits ( a = b = c ) (specifically, when all are zero). The other conditions force ( a ), ( b ), and ( c ) to be distinct, which violates the necessary condition for the equation to hold.So, after carefully going through each option and verifying, I conclude that the correct answer is Option E.

💡Okay, so I've got this problem here about Kevin planting corn and cotton. He has 30 pounds of corn and some amount of cotton, which we're supposed to find. The corn sells for 5 per pound and the cotton for 10 per pound. When he sells all of them, he makes a total of 640. Alright, let's break this down. First, I know that the total money he makes is from both corn and cotton. So, if I can figure out how much money he makes from each, I can find out how much cotton he sold.He has 30 pounds of corn, and each pound sells for 5. So, the revenue from corn should be 30 times 5. Let me calculate that: 30 * 5 = 150. So, he makes 150 from corn.Now, the total revenue is 640, and he already made 150 from corn. That means the rest must come from cotton. So, subtracting the corn revenue from the total gives me the cotton revenue: 640 - 150 = 490. So, he made 490 from cotton.Since cotton sells for 10 per pound, I can find out how many pounds he sold by dividing the total cotton revenue by the price per pound. So, 490 divided by 10 is 49. That means he sold 49 pounds of cotton.Wait, but the problem is asking for x, which is the pounds of cotton. So, x should be 49. Hmm, but I just want to make sure I didn't make any mistakes in my calculations.Let me double-check. 30 pounds of corn at 5 per pound: 30 * 5 = 150. Cotton revenue: 640 - 150 = 490. Cotton sold: 490 / 10 = 49. Yeah, that seems right.So, I think x is 49. But wait, the initial answer given was 8. That doesn't make sense. Did I do something wrong? Let me see.Wait, maybe I misread the problem. Let me check again. It says he has 30 pounds of corn and x pounds of cotton. Corn sells for 5 per pound, cotton for 10 per pound. Total revenue is 640. So, 5*30 + 10*x = 640. That's 150 + 10x = 640. So, 10x = 490, x = 49. Hmm, that's what I got.But the initial answer was 8. Maybe there's a mistake in the initial answer. Or perhaps I misread the problem. Let me check the problem statement again."Kevin plants corn and cotton. Once he harvests the crops, he has 30 pounds of corn and x pounds of cotton. Corn sells for 5 per pound and cotton sells for 10 per pound. If Kevin sells all his corn and cotton for a total of 640, then compute x."No, that seems correct. So, unless there's a typo in the problem, the correct answer should be 49. But the initial answer was 8. Maybe the initial answer was wrong. Or maybe I made a mistake somewhere.Wait, perhaps the problem was about the total weight? No, it's about the total revenue. So, 30 pounds of corn at 5 is 150, and x pounds of cotton at 10 is 10x. Total is 150 + 10x = 640. So, x = 49. Yeah, that seems right.I think the initial answer might have been incorrect. So, I'm confident that x is 49.

💡Okay, so I've got this equation: (x^4 + ax^3 + bx^2 + ax + 1 = 0), and I need to find the least possible value of (a^2 + b^2) given that there's at least one real solution. Hmm, quartic equations can be tricky, but maybe there's a way to simplify this.First, I notice that the equation is palindromic. That means the coefficients read the same forwards and backwards. For a quartic equation, that usually means (x^4 + ax^3 + bx^2 + ax + 1 = 0). I remember that palindromic polynomials have some nice properties. Maybe I can factor this equation somehow.Let me try dividing both sides by (x^2) to make it symmetric. So, dividing by (x^2), I get:[x^2 + ax + b + frac{a}{x} + frac{1}{x^2} = 0]Hmm, that looks better. Now, if I let (y = x + frac{1}{x}), then (y^2 = x^2 + 2 + frac{1}{x^2}), which means (x^2 + frac{1}{x^2} = y^2 - 2). That's useful because I can substitute that into the equation.So substituting, the equation becomes:[(y^2 - 2) + aleft(x + frac{1}{x}right) + b = 0]But (x + frac{1}{x}) is just (y), so this simplifies to:[y^2 - 2 + ay + b = 0]Which is:[y^2 + ay + (b - 2) = 0]Alright, so now I have a quadratic equation in terms of (y). For the original equation to have a real solution, (y) must be real. But also, since (y = x + frac{1}{x}), (y) must satisfy certain conditions. Specifically, for real (x), (y) must be greater than or equal to 2 or less than or equal to -2. This is because by the AM-GM inequality, (x + frac{1}{x} geq 2) if (x > 0), and similarly, if (x < 0), (x + frac{1}{x} leq -2).So, for the quadratic equation (y^2 + ay + (b - 2) = 0) to have a real solution (y) such that (|y| geq 2), the quadratic must have at least one root with absolute value at least 2.Let me denote the roots of the quadratic equation as (y_1) and (y_2). Then, the quadratic can be written as:[(y - y_1)(y - y_2) = 0]Expanding this, we get:[y^2 - (y_1 + y_2)y + y_1 y_2 = 0]Comparing this with our equation (y^2 + ay + (b - 2) = 0), we see that:[-(y_1 + y_2) = a quad Rightarrow quad y_1 + y_2 = -a][y_1 y_2 = b - 2]So, we have relationships between the roots and the coefficients (a) and (b).Now, since we need at least one of the roots (y_1) or (y_2) to satisfy (|y| geq 2), let's consider the conditions for that.Case 1: Suppose (y_1 geq 2). Then, since (y_1 + y_2 = -a), we can express (y_2 = -a - y_1). Similarly, (y_1 y_2 = b - 2).Case 2: Suppose (y_1 leq -2). Similarly, (y_2 = -a - y_1), and (y_1 y_2 = b - 2).But perhaps a better approach is to use the quadratic formula to express the roots and then impose the condition that at least one root has absolute value at least 2.The quadratic equation is (y^2 + ay + (b - 2) = 0), so the roots are:[y = frac{-a pm sqrt{a^2 - 4(b - 2)}}{2}]For the roots to be real, the discriminant must be non-negative:[a^2 - 4(b - 2) geq 0 quad Rightarrow quad a^2 geq 4b - 8]But we also need at least one root to satisfy (|y| geq 2). So, let's consider the roots:[y_1 = frac{-a + sqrt{a^2 - 4(b - 2)}}{2}][y_2 = frac{-a - sqrt{a^2 - 4(b - 2)}}{2}]We need either (y_1 geq 2) or (y_2 leq -2).Let's first consider (y_1 geq 2):[frac{-a + sqrt{a^2 - 4(b - 2)}}{2} geq 2]Multiply both sides by 2:[-a + sqrt{a^2 - 4(b - 2)} geq 4]Rearrange:[sqrt{a^2 - 4(b - 2)} geq a + 4]Now, since the square root is always non-negative, the right-hand side must also be non-negative:[a + 4 geq 0 quad Rightarrow quad a geq -4]Now, square both sides:[a^2 - 4(b - 2) geq (a + 4)^2]Expand the right-hand side:[a^2 - 4b + 8 geq a^2 + 8a + 16]Simplify:[-4b + 8 geq 8a + 16][-4b geq 8a + 8]Divide both sides by -4 (remembering to reverse the inequality sign):[b leq -2a - 2]Okay, so that's one condition: (b leq -2a - 2).Now, let's consider the other case where (y_2 leq -2):[frac{-a - sqrt{a^2 - 4(b - 2)}}{2} leq -2]Multiply both sides by 2:[-a - sqrt{a^2 - 4(b - 2)} leq -4]Rearrange:[-sqrt{a^2 - 4(b - 2)} leq -4 + a]Multiply both sides by -1 (reversing the inequality):[sqrt{a^2 - 4(b - 2)} geq 4 - a]Again, since the square root is non-negative, the right-hand side must also be non-negative:[4 - a geq 0 quad Rightarrow quad a leq 4]Now, square both sides:[a^2 - 4(b - 2) geq (4 - a)^2]Expand the right-hand side:[a^2 - 4b + 8 geq 16 - 8a + a^2]Simplify:[-4b + 8 geq 16 - 8a][-4b geq 8 - 8a]Divide both sides by -4 (reversing the inequality):[b leq -2 + 2a]So, the second condition is (b leq 2a - 2).Therefore, combining both cases, we have two inequalities:1. (b leq -2a - 2)2. (b leq 2a - 2)So, depending on the value of (a), either one or both of these inequalities will hold.Now, our goal is to minimize (a^2 + b^2) subject to these inequalities. So, we can think of this as an optimization problem where we need to find the minimum of (a^2 + b^2) given that (b leq -2a - 2) or (b leq 2a - 2).Let me visualize this. The region defined by (b leq -2a - 2) is a half-plane below the line (b = -2a - 2), and similarly, (b leq 2a - 2) is a half-plane below the line (b = 2a - 2). The feasible region is the union of these two half-planes.We need to find the point ((a, b)) in this feasible region that is closest to the origin, since (a^2 + b^2) is the squared distance from the origin.The closest point to the origin on a line (b = ma + c) is given by the formula:[text{Distance} = frac{|c|}{sqrt{1 + m^2}}]But since we have inequalities, we need to check which line gives the smaller distance.First, let's consider the line (b = -2a - 2). The distance from the origin to this line is:[frac{| -2 |}{sqrt{1 + (-2)^2}} = frac{2}{sqrt{5}} approx 0.894]Similarly, for the line (b = 2a - 2), the distance is:[frac{| -2 |}{sqrt{1 + (2)^2}} = frac{2}{sqrt{5}} approx 0.894]So both lines are equidistant from the origin. Therefore, the minimal (a^2 + b^2) will be the square of this distance, which is (left(frac{2}{sqrt{5}}right)^2 = frac{4}{5}).But wait, we need to ensure that the closest point lies within the feasible region. Since the feasible region is the union of the regions below both lines, the closest point to the origin on either line will lie within the feasible region because the lines are below the origin.Therefore, the minimal value of (a^2 + b^2) is (frac{4}{5}).To confirm, let's find the exact point where the distance is minimized. For the line (b = -2a - 2), the closest point is given by:[a = frac{-2 cdot (-2)}{1 + (-2)^2} = frac{4}{5}]Wait, no, that's not the correct formula. Let me recall the formula for the projection of a point onto a line.Given a line (Ax + By + C = 0), the projection of the origin ((0,0)) onto this line is:[left( frac{-AC}{A^2 + B^2}, frac{-BC}{A^2 + B^2} right)]So, for the line (b = -2a - 2), we can rewrite it as (2a + b + 2 = 0). So, (A = 2), (B = 1), (C = 2).Wait, no, actually, the standard form is (Ax + By + C = 0). So, (2a + b + 2 = 0) is correct.Thus, the projection is:[a = frac{-A C}{A^2 + B^2} = frac{-2 cdot 2}{4 + 1} = frac{-4}{5}][b = frac{-B C}{A^2 + B^2} = frac{-1 cdot 2}{5} = frac{-2}{5}]So, the closest point on the line (2a + b + 2 = 0) is (left( -frac{4}{5}, -frac{2}{5} right)).Calculating (a^2 + b^2):[left( -frac{4}{5} right)^2 + left( -frac{2}{5} right)^2 = frac{16}{25} + frac{4}{25} = frac{20}{25} = frac{4}{5}]Similarly, for the line (b = 2a - 2), which can be rewritten as (-2a + b + 2 = 0), so (A = -2), (B = 1), (C = 2).The projection is:[a = frac{-A C}{A^2 + B^2} = frac{-(-2) cdot 2}{4 + 1} = frac{4}{5}][b = frac{-B C}{A^2 + B^2} = frac{-1 cdot 2}{5} = frac{-2}{5}]So, the closest point is (left( frac{4}{5}, -frac{2}{5} right)), and (a^2 + b^2) is also (frac{16}{25} + frac{4}{25} = frac{20}{25} = frac{4}{5}).Therefore, both lines give the same minimal value of (frac{4}{5}), and the points are symmetric with respect to the y-axis.Hence, the least possible value of (a^2 + b^2) is (frac{4}{5}).