💡Okay, so I have this problem here: "A rectangular hyperbola with coordinate axes as its axes of symmetry passes through the point (2, √2). The equation of this hyperbola is __________." Hmm, I need to find the equation of this hyperbola. Let me think about how to approach this.First, I remember that a hyperbola is a type of conic section, and it has two axes of symmetry. In this case, the coordinate axes (the x-axis and y-axis) are the axes of symmetry for the hyperbola. That probably means the hyperbola is centered at the origin, right? Because the coordinate axes intersect at the origin, so if they are the axes of symmetry, the center must be there.Now, I recall that the standard equation of a hyperbola centered at the origin can be either of the form (x²/a²) - (y²/b²) = 1 or (y²/b²) - (x²/a²) = 1, depending on whether it opens horizontally or vertically. Since the problem mentions it's a rectangular hyperbola, I think that has some specific properties.Wait, what's a rectangular hyperbola? I think it's a hyperbola where the asymptotes are perpendicular to each other. For a standard hyperbola, the asymptotes have slopes of ±(b/a) or ±(a/b), depending on the orientation. So, for them to be perpendicular, the product of their slopes should be -1. That means (b/a) * (a/b) = 1, but that's not -1. Hmm, maybe I'm mixing something up.Oh, wait! For a rectangular hyperbola, the asymptotes are perpendicular, so the slopes should multiply to -1. Let me think: if the hyperbola is of the form (x²/a²) - (y²/b²) = 1, then the slopes of the asymptotes are ±(b/a). For these to be perpendicular, the product of the slopes should be -1. So, (b/a) * (-b/a) = - (b²/a²) = -1. Therefore, (b²/a²) = 1, which implies that b² = a². So, b = a. That means in a rectangular hyperbola, the transverse and conjugate axes are equal in length.So, if b = a, then the equation simplifies. Let me write that down. If b = a, then the equation becomes (x²/a²) - (y²/a²) = 1, which can be written as (x² - y²)/a² = 1, or x² - y² = a². So, the equation is x² - y² = a². Alternatively, sometimes it's written as xy = c² for a rectangular hyperbola rotated by 45 degrees, but in this case, since the axes of symmetry are the coordinate axes, it's not rotated. So, I think x² - y² = a² is the correct form.Now, the hyperbola passes through the point (2, √2). So, I can substitute x = 2 and y = √2 into the equation to find the value of a².Let me do that:x² - y² = a²(2)² - (√2)² = a²4 - 2 = a²2 = a²So, a² is 2. Therefore, the equation of the hyperbola is x² - y² = 2.Wait, let me double-check that. If I plug in x = 2 and y = √2 into x² - y², I get 4 - 2, which is 2, which equals a². So, that seems correct.But just to make sure I didn't make any mistakes, let me go through the steps again.1. The hyperbola is rectangular, so it has perpendicular asymptotes. That means the slopes of the asymptotes multiply to -1.2. For a hyperbola centered at the origin with axes along the coordinate axes, the equation is (x²/a²) - (y²/b²) = 1.3. The slopes of the asymptotes are ±(b/a). For them to be perpendicular, (b/a) * (-b/a) = -1, which simplifies to (b²/a²) = 1, so b = a.4. Substituting b = a into the equation gives x² - y² = a².5. The hyperbola passes through (2, √2), so plugging in x = 2 and y = √2 gives 4 - 2 = 2, so a² = 2.6. Therefore, the equation is x² - y² = 2.That all seems to check out. I don't think I made any mistakes here. So, I'm confident that the equation of the hyperbola is x² - y² = 2.Just to visualize it, this hyperbola opens to the left and right along the x-axis because the positive term is with x². The asymptotes would be the lines y = ±x, since the slopes are ±(b/a) and b = a. So, the asymptotes are y = x and y = -x, which are indeed perpendicular to each other, confirming that it's a rectangular hyperbola.Also, since the hyperbola passes through (2, √2), which is in the first quadrant, and given the equation x² - y² = 2, plugging in x = 2 and y = √2 gives 4 - 2 = 2, which satisfies the equation. So, that point lies on the hyperbola.I think that's all. I can't think of any other steps or considerations needed here. The key was recognizing that it's a rectangular hyperbola, which tells us that a = b, simplifying the equation, and then using the given point to find the specific value of a².So, yeah, the equation is x² - y² = 2.**Final Answer**The equation of the hyperbola is boxed{x^2 - y^2 = 2}.

💡Okay, so I have this problem where I need to find the sum of the roots of a quadratic trinomial ( P(x) ). The condition given is that for all real numbers ( x ), the inequality ( P(x^3 + x) geq P(x^2 + 1) ) holds. Hmm, that's interesting. Let me try to break this down step by step.First, I know that a quadratic trinomial generally looks like ( P(x) = ax^2 + bx + c ), where ( a ), ( b ), and ( c ) are real coefficients, and ( a ) is not zero. The sum of the roots of ( P(x) ) can be found using the formula ( -frac{b}{a} ). So, if I can find the relationship between ( a ) and ( b ), I can determine the sum of the roots.The key condition here is that ( P(x^3 + x) geq P(x^2 + 1) ) for all real ( x ). Let me define a new function ( f(x) = P(x^3 + x) - P(x^2 + 1) ). Since ( P(x^3 + x) geq P(x^2 + 1) ) for all ( x ), this means that ( f(x) geq 0 ) for all ( x ).Now, if ( f(x) ) is always non-negative, it must attain its minimum value of zero at some point. Let me check what happens when ( x = 1 ). Plugging in, I get:( f(1) = P(1^3 + 1) - P(1^2 + 1) = P(2) - P(2) = 0 ).So, ( f(1) = 0 ). That means the function ( f(x) ) has a minimum at ( x = 1 ). Therefore, the derivative of ( f(x) ) at ( x = 1 ) should be zero. Let's compute ( f'(x) ).First, find the derivative of ( f(x) ):( f'(x) = frac{d}{dx}[P(x^3 + x) - P(x^2 + 1)] ).Using the chain rule, this becomes:( f'(x) = P'(x^3 + x) cdot (3x^2 + 1) - P'(x^2 + 1) cdot (2x) ).Now, evaluate this at ( x = 1 ):( f'(1) = P'(1^3 + 1) cdot (3 cdot 1^2 + 1) - P'(1^2 + 1) cdot (2 cdot 1) ).Simplifying, we get:( f'(1) = P'(2) cdot 4 - P'(2) cdot 2 ).Since ( f'(1) = 0 ), this simplifies to:( 4P'(2) - 2P'(2) = 0 implies 2P'(2) = 0 implies P'(2) = 0 ).Okay, so the derivative of ( P(x) ) at ( x = 2 ) is zero. Let's find ( P'(x) ). Since ( P(x) = ax^2 + bx + c ), the derivative is:( P'(x) = 2ax + b ).Setting ( P'(2) = 0 ):( 2a cdot 2 + b = 0 implies 4a + b = 0 implies b = -4a ).Great, so now we have ( b = -4a ). The sum of the roots of ( P(x) ) is given by ( -frac{b}{a} ). Substituting ( b = -4a ):Sum of the roots = ( -frac{-4a}{a} = 4 ).So, the sum of the roots of ( P(x) ) is 4. That seems straightforward, but let me double-check to make sure I didn't miss anything.Wait, I assumed that ( f(x) ) has a minimum at ( x = 1 ) because ( f(1) = 0 ) and ( f(x) geq 0 ) everywhere. Is there a possibility that ( f(x) ) could have multiple minima or other critical points? Well, ( f(x) ) is a composition of polynomials, so it's smooth and differentiable everywhere. Since ( f(x) geq 0 ) for all ( x ), the minimum at ( x = 1 ) is a global minimum. Therefore, the derivative at that point must indeed be zero, which gives us the necessary condition ( P'(2) = 0 ). I think that covers it. The key steps were recognizing the minimum at ( x = 1 ), computing the derivative, and using that to find the relationship between ( a ) and ( b ). Then, applying the formula for the sum of the roots gives the answer. I don't see any gaps in this reasoning, so I feel confident that the sum of the roots is 4.**Final Answer**boxed{4}

💡Okay, so I have this equation: (x^{10} + 8x^9 + 16x^8 + 2023x^7 - 2023x^6 = 0). I need to figure out how many positive real solutions exist for this equation. Hmm, let me think about how to approach this.First, I remember that when dealing with polynomial equations, factoring can sometimes help simplify the problem. Let me see if I can factor out any common terms from this equation. Looking at each term, I notice that each has a factor of (x^6). So, I can factor that out:(x^6(x^4 + 8x^3 + 16x^2 + 2023x - 2023) = 0).Alright, so now I have two factors multiplied together: (x^6) and another polynomial of degree 4. Since we're looking for positive real solutions, (x = 0) isn't a positive solution, so I can focus on the quartic polynomial:(x^4 + 8x^3 + 16x^2 + 2023x - 2023 = 0).Now, I need to find how many positive real roots this quartic equation has. I remember that Descartes' Rule of Signs can help determine the number of positive real roots by looking at the sign changes in the polynomial's coefficients.Let me list the coefficients: 1 (for (x^4)), 8 (for (x^3)), 16 (for (x^2)), 2023 (for (x)), and -2023 (constant term). Looking at the signs: all the coefficients except the last one are positive. The last term is negative. So, how many sign changes are there? From positive to positive, positive to positive, positive to positive, and then positive to negative. That's just one sign change. According to Descartes' Rule of Signs, the number of positive real roots is either equal to the number of sign changes or less than it by an even number. Since there's only one sign change, that means there is exactly one positive real root.Wait, but I should double-check this. Maybe I can use the Intermediate Value Theorem to confirm. Let me evaluate the quartic polynomial at some points to see where it crosses the x-axis.Let's compute (f(0)):(f(0) = 0^4 + 8*0^3 + 16*0^2 + 2023*0 - 2023 = -2023).So, (f(0) = -2023), which is negative.Now, let's compute (f(1)):(f(1) = 1 + 8 + 16 + 2023 - 2023 = 1 + 8 + 16 + 0 = 25).So, (f(1) = 25), which is positive.Since the function changes from negative at (x=0) to positive at (x=1), by the Intermediate Value Theorem, there must be at least one root between 0 and 1.But Descartes' Rule told me there's exactly one positive real root, so that must be the only one. Just to be thorough, let me check another point. How about (x=2):(f(2) = 16 + 64 + 64 + 4046 - 2023 = 16 + 64 + 64 + 4046 - 2023).Calculating step by step:16 + 64 = 8080 + 64 = 144144 + 4046 = 41904190 - 2023 = 2167.So, (f(2) = 2167), which is still positive. What about a larger value, say (x=10):(f(10) = 10000 + 8000 + 1600 + 20230 - 2023).Calculating:10000 + 8000 = 1800018000 + 1600 = 1960019600 + 20230 = 3983039830 - 2023 = 37807.So, (f(10) = 37807), which is positive as well.Since the polynomial is positive at (x=1), (x=2), and (x=10), and it only crosses from negative to positive once between 0 and 1, there's no indication of any other roots beyond that. I can also think about the behavior of the polynomial as (x) approaches infinity. The leading term is (x^4), which dominates, so as (x) becomes very large, (f(x)) tends to positive infinity. Similarly, as (x) approaches negative infinity, (x^4) is still positive, but since we're only concerned with positive real solutions, that's not relevant here.Another way to think about it is to consider the derivative of the quartic polynomial to analyze its increasing or decreasing behavior. Let's compute (f'(x)):(f'(x) = 4x^3 + 24x^2 + 32x + 2023).All the coefficients of (f'(x)) are positive, which means (f'(x)) is always positive for (x > 0). Therefore, the function (f(x)) is strictly increasing for all positive (x). Since it's strictly increasing and it crosses the x-axis from negative to positive exactly once between 0 and 1, there can only be one positive real root.So, putting it all together, there is exactly one positive real solution to the original equation.

💡Okay, so I have this problem to rewrite the proposition "The acute angles of a right triangle are complementary" in the form "If ..., then ...". Hmm, let's see. First, I need to understand what each part of the original statement means. A right triangle is a triangle that has one right angle, which is 90 degrees. The other two angles are called acute angles because they are less than 90 degrees. Now, complementary angles are two angles that add up to 90 degrees. So, the original statement is saying that in a right triangle, the two acute angles add up to 90 degrees.Alright, so I need to rephrase this in the form of an "If ..., then ..." statement. The structure of such a statement is: If [condition], then [result]. So, I need to identify what the condition is and what the result is in the original proposition.The condition here is that we have a right triangle. The result is that the acute angles are complementary. So, putting it together, it would be: If a triangle is a right triangle, then its acute angles are complementary.Wait, let me make sure I'm not missing anything. Is there any other way to interpret the original statement? Could it be referring to something else? I don't think so. It's pretty straightforward. The key elements are the right triangle and the complementary acute angles.Maybe I should double-check the definitions to be certain. A right triangle has one 90-degree angle, and the other two angles must add up to 90 degrees because the total sum of angles in any triangle is 180 degrees. So, yes, the two acute angles must be complementary.Is there a more precise way to phrase it? Perhaps specifying that it's about the acute angles. So, "If a triangle is a right triangle, then its acute angles are complementary." That seems clear and accurate.I think that's the correct way to rephrase the proposition in the desired form. It clearly states the condition (being a right triangle) and the result (acute angles being complementary).

💡Alright, so I've got this problem here: the function f(x) = a^x, where a is greater than 0 and not equal to 1, has its maximum value larger than its minimum value by a/2 on the interval [1, 2]. I need to find the value of a. Hmm, okay, let me think about how to approach this.First, I remember that exponential functions like a^x can be either increasing or decreasing depending on the value of a. If a is greater than 1, the function is increasing, and if a is between 0 and 1, the function is decreasing. So, I think I need to consider both cases separately.Let me start with the case where a > 1. If a is greater than 1, then as x increases, a^x increases as well. That means on the interval [1, 2], the function will have its minimum value at x = 1 and its maximum value at x = 2. So, the minimum value is f(1) = a^1 = a, and the maximum value is f(2) = a^2.The problem states that the maximum is larger than the minimum by a/2. So, the difference between the maximum and minimum should be a/2. That gives me the equation:a^2 - a = a/2Hmm, let me write that down:a^2 - a = (a)/2To solve this, I can subtract (a)/2 from both sides to get:a^2 - a - (a)/2 = 0Combining like terms:a^2 - (3a)/2 = 0I can factor out an a:a(a - 3/2) = 0So, the solutions are a = 0 or a = 3/2. But since a > 1, a = 0 doesn't make sense here. So, a = 3/2 is a possible solution.Okay, that was the case when a > 1. Now, let's consider the other case where 0 < a < 1. In this scenario, the function a^x is decreasing. So, on the interval [1, 2], the maximum value will be at x = 1, and the minimum value will be at x = 2. Therefore, the maximum is f(1) = a, and the minimum is f(2) = a^2.Again, the problem says the maximum is larger than the minimum by a/2. So, the difference should be:a - a^2 = a/2Let me write that equation:a - a^2 = (a)/2Subtracting (a)/2 from both sides:a - a^2 - (a)/2 = 0Combining like terms:( a - (a)/2 ) - a^2 = 0Which simplifies to:(a)/2 - a^2 = 0Factor out an a:a(1/2 - a) = 0So, the solutions are a = 0 or a = 1/2. Since we're considering 0 < a < 1, a = 0 is invalid, so a = 1/2 is another possible solution.Wait, so I have two solutions: a = 3/2 and a = 1/2. Let me verify both to make sure they fit the original condition.First, let's check a = 3/2. For a = 3/2, f(1) = 3/2 and f(2) = (3/2)^2 = 9/4. The difference is 9/4 - 3/2 = 9/4 - 6/4 = 3/4. Now, a/2 is (3/2)/2 = 3/4. So, the difference is indeed equal to a/2. That checks out.Now, let's check a = 1/2. For a = 1/2, f(1) = 1/2 and f(2) = (1/2)^2 = 1/4. The difference is 1/2 - 1/4 = 1/4. Now, a/2 is (1/2)/2 = 1/4. Again, the difference is equal to a/2. That also checks out.So, both values of a satisfy the given condition. Therefore, the possible values of a are 3/2 and 1/2.Wait, but the problem says "the function... has its maximum value larger than its minimum value by a/2." It doesn't specify whether a is greater than 1 or less than 1, just that a > 0 and a ≠ 1. So, both solutions are valid.Hmm, but the problem asks to "find the value of a," implying maybe a single answer. But since both 3/2 and 1/2 satisfy the condition, I think both are correct. So, I should present both solutions.Let me recap:1. For a > 1: - Maximum at x = 2: f(2) = a^2 - Minimum at x = 1: f(1) = a - Difference: a^2 - a = a/2 - Solution: a = 3/22. For 0 < a < 1: - Maximum at x = 1: f(1) = a - Minimum at x = 2: f(2) = a^2 - Difference: a - a^2 = a/2 - Solution: a = 1/2Both solutions are valid because they satisfy the original condition, so the possible values of a are 3/2 and 1/2.I think that's it. I don't see any mistakes in my calculations, and both solutions check out when I plug them back into the original problem. So, I'm confident that these are the correct values of a.**Final Answer**The possible values of ( a ) are boxed{dfrac{3}{2}} and boxed{dfrac{1}{2}}.

💡Okay, so I have this problem where I'm given ( y = frac{1 + isqrt{3}}{2} ) and I need to calculate ( frac{1}{y^3 - y} ). Hmm, let's see. I remember that ( i ) is the imaginary unit, where ( i = sqrt{-1} ). First, I think I should compute ( y^3 ) and then subtract ( y ) from it. Once I have ( y^3 - y ), I can find its reciprocal. But maybe there's a smarter way to do this without having to compute all the powers directly. Let me think.Wait, ( y ) looks familiar. It resembles the complex cube roots of 1. The cube roots of 1 are 1, ( frac{-1 + isqrt{3}}{2} ), and ( frac{-1 - isqrt{3}}{2} ). Oh, actually, ( y ) is one of them, but it's ( frac{1 + isqrt{3}}{2} ). Hmm, maybe that's a primitive cube root of 1? Let me check.If I cube ( y ), does it equal 1? Let's compute ( y^3 ). First, let's compute ( y^2 ):( y = frac{1 + isqrt{3}}{2} )So, ( y^2 = left( frac{1 + isqrt{3}}{2} right)^2 )Expanding that, it's ( frac{1 + 2isqrt{3} + (isqrt{3})^2}{4} )Simplify ( (isqrt{3})^2 = i^2 * 3 = -3 )So, ( y^2 = frac{1 + 2isqrt{3} - 3}{4} = frac{-2 + 2isqrt{3}}{4} = frac{-1 + isqrt{3}}{2} )Okay, so ( y^2 = frac{-1 + isqrt{3}}{2} ). Now, let's compute ( y^3 ):( y^3 = y cdot y^2 = frac{1 + isqrt{3}}{2} cdot frac{-1 + isqrt{3}}{2} )Multiplying the numerators: ( (1)(-1) + (1)(isqrt{3}) + (isqrt{3})(-1) + (isqrt{3})(isqrt{3}) )That's ( -1 + isqrt{3} - isqrt{3} + (i^2 * 3) )Simplify: ( -1 + 0 + (-3) = -4 )So, ( y^3 = frac{-4}{4} = -1 )Wait, so ( y^3 = -1 ). Interesting. So, ( y^3 - y = -1 - y ). Let's compute that:( y^3 - y = -1 - frac{1 + isqrt{3}}{2} = -frac{2}{2} - frac{1 + isqrt{3}}{2} = -frac{3 + isqrt{3}}{2} )So, ( y^3 - y = -frac{3 + isqrt{3}}{2} ). Now, I need to find the reciprocal of this, which is ( frac{1}{y^3 - y} = frac{1}{ -frac{3 + isqrt{3}}{2} } = -frac{2}{3 + isqrt{3}} )To simplify ( frac{2}{3 + isqrt{3}} ), I can multiply the numerator and denominator by the complex conjugate of the denominator, which is ( 3 - isqrt{3} ):( frac{2}{3 + isqrt{3}} times frac{3 - isqrt{3}}{3 - isqrt{3}} = frac{2(3 - isqrt{3})}{(3)^2 - (isqrt{3})^2} )Simplify the denominator: ( 9 - (i^2 * 3) = 9 - (-3) = 12 )So, the numerator is ( 6 - 2isqrt{3} )Thus, ( frac{6 - 2isqrt{3}}{12} = frac{6}{12} - frac{2isqrt{3}}{12} = frac{1}{2} - frac{isqrt{3}}{6} )But remember, we had a negative sign earlier, so:( -frac{2}{3 + isqrt{3}} = -left( frac{1}{2} - frac{isqrt{3}}{6} right) = -frac{1}{2} + frac{isqrt{3}}{6} )So, the final result is ( -frac{1}{2} + frac{isqrt{3}}{6} ), which corresponds to option (A).Wait, let me double-check my steps to make sure I didn't make a mistake. Starting from ( y^3 = -1 ), that seems correct because when I computed ( y^3 ), it simplified to -1. Then, subtracting y gives ( -1 - y ), which is ( -1 - frac{1 + isqrt{3}}{2} ), leading to ( -frac{3 + isqrt{3}}{2} ). Taking the reciprocal, I multiplied by the conjugate and simplified, ending up with ( -frac{1}{2} + frac{isqrt{3}}{6} ). Yeah, that seems right. I think option (A) is correct.