answer：First, I recognize that the running pace is constant, meaning the time taken is directly proportional to the distance covered.Next, I set up a proportion to relate the time and distance for both the grocery store and my cousin's house.Then, I solve for the unknown time by cross-multiplying and simplifying the equation.Finally, I find that it will take 15 minutes to run to my cousin's house.

answer：First, I need to understand the problem correctly. Carol spends 1/4 of her savings on a stereo and some fraction less than that on a television. The total amount she spends on both the stereo and television is 0.25 of her savings.Let’s denote Carol's total savings as ( S ).She spends ( frac{1}{4} ) of her savings on a stereo, which is:[frac{1}{4} times S]She also spends some fraction on a television, let's call this fraction ( X ). So she spends:[X times S]According to the information given, the total amount she spends on both the stereo and television is 0.25 of her savings:[frac{1}{4} times S + X times S = 0.25 times S]We know that ( frac{1}{4} times S ) is the same as ( 0.25 times S ), so the fraction she spends on the television must be 0. This means she did not spend any of her savings on the television, which is not possible according to the problem statement.There seems to be a misunderstanding. If Carol spent ( frac{1}{4} ) of her savings on the stereo, that is already 0.25 of her savings. If she spent 0.25 of her savings on both the stereo and television combined, then she could not have spent any additional amount on the television.Please check the problem statement again. If Carol spent ( frac{1}{4} ) of her savings on the stereo, and the total spent on both the stereo and television is ( boxed{0.25} ) of her savings, then she did not spend any additional amount on the television. If the problem meant that she spent 0.25 of her savings in addition to the stereo, then the total spent would be more than 0.25 of her savings.

answer：Okay, so I need to find the value of f(2) where the function f(x) is given as x squared minus three x. Alright, let me think about this step by step. First, I remember that when you have a function like f(x) = x² - 3x, and you're asked to find f(2), it means you substitute x with 2 in the function. So, I should replace every x in the equation with the number 2. Let me write that out. So, f(2) would be equal to (2) squared minus three times 2. Hmm, okay, let me calculate each part separately to make sure I don't make any mistakes. First, 2 squared is 4. That part seems straightforward. Then, three times 2 is 6. So now, I have f(2) equals 4 minus 6. Wait, 4 minus 6 is negative 2. So, does that mean f(2) is -2? That seems right, but let me double-check my calculations to be sure. Maybe I should go through it again. Starting with f(x) = x² - 3x, substituting x with 2 gives me f(2) = (2)² - 3*(2). Calculating the exponent first, 2 squared is indeed 4. Then, multiplying 3 by 2 gives me 6. Subtracting 6 from 4 gives me -2. Yep, that checks out. I wonder if there's another way to approach this problem to confirm my answer. Maybe I can graph the function f(x) = x² - 3x and see what the value is at x=2. If I plot the function, it's a parabola opening upwards because the coefficient of x² is positive. The vertex of this parabola can be found using the formula -b/(2a), where a is 1 and b is -3. So, the vertex is at x = 3/(2*1) = 1.5. Plugging x=1.5 into the function gives me f(1.5) = (1.5)² - 3*(1.5) = 2.25 - 4.5 = -2.25. So, the vertex is at (1.5, -2.25). Now, if I move to the right of the vertex by 0.5 units to x=2, the function should increase because the parabola is opening upwards. So, starting from the vertex at x=1.5, which is the minimum point, moving to x=2, the function value should be higher than -2.25. Let me calculate f(2) again: (2)^2 is 4, 3*2 is 6, so 4-6 is -2. That's indeed higher than -2.25, which makes sense because it's moving away from the vertex. Another way to think about this is by factoring the function. Let's see, f(x) = x² - 3x can be factored as x(x - 3). So, f(2) would be 2*(2 - 3) = 2*(-1) = -2. That's the same answer as before. I could also use the concept of limits or derivatives, but that might be overcomplicating things for this problem. Since it's a straightforward substitution, I think my initial method is sufficient. Just to make sure, let me plug in another value to see if my function behaves as expected. For example, f(0) should be 0² - 3*0 = 0. And f(3) would be 9 - 9 = 0. So, the function crosses the x-axis at x=0 and x=3, which aligns with the factored form x(x - 3). If I were to create a table of values for f(x), it would look something like this:x | f(x)---|---0 | 01 | 1 - 3 = -22 | 4 - 6 = -23 | 9 - 9 = 04 | 16 - 12 = 4Looking at this table, at x=2, the function value is indeed -2, which matches my previous calculations. I think I've verified this in multiple ways: substitution, factoring, graphing, and creating a table of values. All methods consistently give me f(2) = -2. I feel confident that this is the correct answer. One last thing I can do is to visualize the graph. Since it's a parabola opening upwards with roots at 0 and 3, the vertex is at (1.5, -2.25). So, at x=2, which is halfway between 1.5 and 3, the function should be increasing from the vertex. The value at x=2 is -2, which is higher than the vertex value of -2.25, confirming that the function is indeed increasing as we move right from the vertex. In summary, substituting x=2 into the function f(x) = x² - 3x gives f(2) = -2. All alternative methods and checks support this result, so I'm certain that f(2) is -2.

answer：Okay, so I have this problem where I need to find the distance from point P(-2,1,2) to a plane α. The normal vector of the plane is given as n = (-2, -2, 1), and there's a point A(-1, 3, 0) that lies on the plane. Hmm, let me think about how to approach this.First, I remember that the distance from a point to a plane can be found using the formula involving the normal vector and a point on the plane. I think the formula is something like the absolute value of the dot product of the vector from the point on the plane to the external point and the normal vector, divided by the magnitude of the normal vector. Let me write that down to make it clearer.So, if I have a point P and a plane with normal vector n, and a point A on the plane, the distance d from P to the plane is:d = |(P - A) · n| / |n|Alright, that seems right. So, I need to find the vector from A to P, which is P - A. Let me compute that.Point A is (-1, 3, 0) and point P is (-2, 1, 2). So, subtracting A from P:P - A = (-2 - (-1), 1 - 3, 2 - 0) = (-2 + 1, 1 - 3, 2 - 0) = (-1, -2, 2)Okay, so the vector AP is (-1, -2, 2). Now, I need to take the dot product of this vector with the normal vector n, which is (-2, -2, 1).Dot product = (-1)*(-2) + (-2)*(-2) + (2)*(1) = 2 + 4 + 2 = 8Hmm, that's positive 8. Now, I need the magnitude of the normal vector n. The normal vector is (-2, -2, 1), so its magnitude is:|n| = sqrt((-2)^2 + (-2)^2 + (1)^2) = sqrt(4 + 4 + 1) = sqrt(9) = 3So, the magnitude is 3. Now, plug these into the distance formula:d = |8| / 3 = 8/3Wait, that seems straightforward. But let me double-check my steps to make sure I didn't make any mistakes.First, vector AP: P - A = (-2 - (-1), 1 - 3, 2 - 0) = (-1, -2, 2). That looks correct.Dot product of AP and n: (-1)*(-2) + (-2)*(-2) + (2)*(1) = 2 + 4 + 2 = 8. Yep, that's right.Magnitude of n: sqrt(4 + 4 + 1) = sqrt(9) = 3. Correct.So, distance d = 8/3. Hmm, 8 divided by 3 is approximately 2.666..., but since they probably want the exact value, 8/3 is fine.Wait, just to make sure, is there another way to approach this? Maybe by finding the equation of the plane first and then using the standard distance formula?Let me try that as a verification. The general equation of a plane is given by:n · (X - A) = 0Where n is the normal vector, and A is a point on the plane. So, plugging in the values:(-2, -2, 1) · (x - (-1), y - 3, z - 0) = 0Simplify that:-2(x + 1) - 2(y - 3) + 1(z - 0) = 0Expanding:-2x - 2 - 2y + 6 + z = 0Combine like terms:-2x - 2y + z + (-2 + 6) = 0Which simplifies to:-2x - 2y + z + 4 = 0So, the equation of the plane is -2x - 2y + z + 4 = 0.Now, the standard distance formula from a point (x0, y0, z0) to the plane ax + by + cz + d = 0 is:d = |a*x0 + b*y0 + c*z0 + d| / sqrt(a^2 + b^2 + c^2)So, in this case, a = -2, b = -2, c = 1, d = 4, and the point P is (-2, 1, 2).Plugging into the formula:d = |-2*(-2) + (-2)*1 + 1*2 + 4| / sqrt((-2)^2 + (-2)^2 + 1^2)Calculate numerator:-2*(-2) = 4(-2)*1 = -21*2 = 2So, adding these up: 4 - 2 + 2 + 4 = 8Denominator is sqrt(4 + 4 + 1) = sqrt(9) = 3So, distance d = |8| / 3 = 8/3Same result as before. Okay, that confirms it. So, the distance is indeed 8/3.I think I've got it right. Both methods give the same answer, so I'm confident that the distance from point P to plane α is 8/3 units.**Final Answer**The distance from point ( P(-2,1,2) ) to plane ( alpha ) is boxed{dfrac{8}{3}}.

answer：Okay, so I have this problem about a regular dodecagon, which is a 12-sided polygon, right? It's labeled Q1 through Q12, and it's drawn on the coordinate plane. They've given me specific points: Q1 is at (4,0) and Q7 is at (2,0). I need to compute the product of all the complex numbers corresponding to each vertex of the dodecagon. That is, I need to find the product (x1 + y1 i)(x2 + y2 i)...(x12 + y12 i).Hmm, okay. So, first, I know that a regular dodecagon has all sides equal and all internal angles equal. Since it's regular, it should be symmetric, and all the vertices lie on a circle. So, I can model this dodecagon as being inscribed in a circle in the complex plane.Each vertex Qn can be represented as a complex number zn = xn + yn i. So, the product I need is z1*z2*z3*...*z12.Now, I remember that when dealing with roots of unity, the product of all the roots is related to the constant term of the polynomial. But in this case, it's not exactly the roots of unity because the dodecagon isn't centered at the origin. Q1 is at (4,0) and Q7 is at (2,0). So, the center isn't at (0,0). Let me figure out where the center is.Since it's a regular dodecagon, the center should be the midpoint between Q1 and Q7 because Q7 is directly opposite Q1 in a regular polygon. So, the midpoint between (4,0) and (2,0) is ((4+2)/2, (0+0)/2) = (3,0). So, the center of the dodecagon is at (3,0).Alright, so the dodecagon is centered at (3,0). That means each vertex can be represented as a complex number shifted by 3 units along the real axis. So, if I denote the center as c = 3 + 0i, then each vertex zn can be written as c + r*e^(iθn), where r is the radius of the circumscribed circle, and θn is the angle for each vertex.Wait, but actually, since it's a regular dodecagon, the angle between each vertex is 360/12 = 30 degrees, or π/6 radians. So, starting from Q1 at (4,0), which is 4 units from the origin, but the center is at (3,0). So, the radius r is the distance from the center to any vertex. Let me compute that.Q1 is at (4,0), and the center is at (3,0). So, the distance between them is |4 - 3| = 1. So, the radius r is 1. That means each vertex is 1 unit away from the center (3,0). So, the complex number for each vertex is 3 + e^(iθn), where θn is the angle for each vertex.But wait, let me double-check. If the radius is 1, then the distance from the center (3,0) to Q1 (4,0) is indeed 1. Similarly, the distance from the center to Q7 (2,0) is also 1. So, that makes sense.So, each vertex zn can be written as 3 + e^(iθn). But actually, since the center is at (3,0), the complex number is 3 + e^(iθn). But wait, in complex plane terms, the center is 3 + 0i, so each vertex is 3 + e^(iθn). But actually, no, that's not quite right. Because if you have a circle centered at (3,0) with radius 1, then each point on the circle is 3 + e^(iθn). But wait, that would place the point at (3 + cosθn, sinθn). But Q1 is at (4,0), which would correspond to θn = 0, so 3 + e^(i0) = 3 + 1 = 4, which is correct. Similarly, Q7 is at (2,0), which would correspond to θn = π, so 3 + e^(iπ) = 3 - 1 = 2, which is also correct. So, yes, each vertex is 3 + e^(iθn), where θn = (n - 1)*π/6, since there are 12 vertices, each separated by 30 degrees or π/6 radians.Wait, hold on. Let me make sure about the angles. Since Q1 is at (4,0), which is θ = 0, then Q2 would be at θ = π/6, Q3 at θ = 2π/6, and so on, up to Q12 at θ = 11π/6. So, yes, θn = (n - 1)*π/6. So, zn = 3 + e^(iθn).But actually, in complex numbers, the center is at 3, so each vertex is 3 + e^(iθn). So, the complex number for each vertex is 3 + e^(iθn). Therefore, the product we're looking for is the product from n=1 to 12 of (3 + e^(iθn)).Wait, but that seems a bit complicated. Maybe there's a better way to approach this. I remember that if you have a polynomial whose roots are the vertices of a regular polygon centered at the origin, then the product of the roots is related to the constant term. But in this case, the center is not at the origin, it's at (3,0). So, maybe I can shift the coordinate system to make the center at the origin.Let me think. If I let w = z - 3, then the center is at 0 in the w-plane. So, each vertex zn in the z-plane corresponds to wn = zn - 3 in the w-plane. So, the product we're looking for is the product from n=1 to 12 of zn = product from n=1 to 12 of (wn + 3).But in the w-plane, the polygon is centered at the origin, and the vertices are the 12th roots of unity scaled by the radius. Wait, but earlier I found that the radius is 1, so the vertices in the w-plane are just the 12th roots of unity. So, wn = e^(iθn) = e^(i(n-1)π/6). So, the product becomes product from n=1 to 12 of (3 + wn), where wn are the 12th roots of unity.So, now, the product is product_{n=1}^{12} (3 + wn). But the 12th roots of unity are the roots of the equation w^12 = 1. So, the polynomial is w^12 - 1 = 0. Therefore, the product product_{n=1}^{12} (w - wn) = w^12 - 1.But in our case, we have product_{n=1}^{12} (3 + wn). Let me see. If I set w = -3, then product_{n=1}^{12} (3 + wn) = product_{n=1}^{12} ( -3 - wn ) = product_{n=1}^{12} ( - (3 + wn) ) = (-1)^12 product_{n=1}^{12} (3 + wn) = product_{n=1}^{12} (3 + wn). But wait, that might not be helpful.Alternatively, since the 12th roots of unity are the roots of w^12 - 1 = 0, then product_{n=1}^{12} (w - wn) = w^12 - 1. So, if I set w = -3, then product_{n=1}^{12} (-3 - wn) = (-3)^12 - 1 = 3^12 - 1. But (-3)^12 is the same as 3^12 because the exponent is even. So, 3^12 is 531441, so 531441 - 1 = 531440.Wait, but product_{n=1}^{12} ( -3 - wn ) = (-1)^12 product_{n=1}^{12} (3 + wn ) = product_{n=1}^{12} (3 + wn ). So, that means product_{n=1}^{12} (3 + wn ) = 3^12 - 1 = 531440.Therefore, the product we're looking for is 531440.Wait, but let me make sure I didn't make a mistake. So, in the w-plane, the product is product_{n=1}^{12} (3 + wn ) = 3^12 - 1. But why is that?Because the polynomial whose roots are wn is w^12 - 1 = 0. So, if I consider the polynomial P(w) = w^12 - 1, then P(-3) = (-3)^12 - 1 = 3^12 - 1. But P(w) = product_{n=1}^{12} (w - wn ). So, P(-3) = product_{n=1}^{12} (-3 - wn ) = (-1)^12 product_{n=1}^{12} (3 + wn ) = product_{n=1}^{12} (3 + wn ). Therefore, product_{n=1}^{12} (3 + wn ) = 3^12 - 1.Yes, that makes sense. So, 3^12 is 531441, so 531441 - 1 = 531440.Therefore, the product of all the zn is 531440.Wait, but let me think again. The zn are the vertices in the z-plane, which are 3 + wn, where wn are the 12th roots of unity. So, the product is product_{n=1}^{12} (3 + wn ). Which we just found is 3^12 - 1.But wait, is that correct? Because the product of (w - wn ) is w^12 - 1, so if I plug in w = -3, I get product_{n=1}^{12} (-3 - wn ) = (-3)^12 - 1 = 3^12 - 1. But (-3 - wn ) = -(3 + wn ), so product_{n=1}^{12} (-3 - wn ) = (-1)^12 product_{n=1}^{12} (3 + wn ) = product_{n=1}^{12} (3 + wn ). Therefore, product_{n=1}^{12} (3 + wn ) = 3^12 - 1.Yes, that seems correct.So, the product is 3^12 - 1. 3^12 is 531441, so 531441 - 1 = 531440.Therefore, the numerical value of the product is 531440.Wait, but let me just make sure I didn't make any miscalculations. 3^1 is 3, 3^2 is 9, 3^3 is 27, 3^4 is 81, 3^5 is 243, 3^6 is 729, 3^7 is 2187, 3^8 is 6561, 3^9 is 19683, 3^10 is 59049, 3^11 is 177147, 3^12 is 531441. Yes, that's correct. So, 531441 - 1 is indeed 531440.So, I think that's the answer.

answer：Okay, so I have this problem here where I need to find the maximum value of the expression (frac{x^2 + a - sqrt{x^4 + a^2}}{x}), where (x) is a positive real number and (a) is a positive constant. Hmm, let me try to figure this out step by step.First, I notice that the expression has a square root in the numerator, which might complicate things. Maybe I can simplify it somehow. I remember that sometimes, when you have an expression with a square root, multiplying by its conjugate can help. The conjugate of (x^2 + a - sqrt{x^4 + a^2}) is (x^2 + a + sqrt{x^4 + a^2}). So, if I multiply both the numerator and the denominator by this conjugate, I might be able to simplify things.Let me write that out:[frac{x^2 + a - sqrt{x^4 + a^2}}{x} times frac{x^2 + a + sqrt{x^4 + a^2}}{x^2 + a + sqrt{x^4 + a^2}}]Multiplying the numerators together, I get:[(x^2 + a)^2 - (sqrt{x^4 + a^2})^2]Simplifying that, the square and the square root cancel out, so it becomes:[(x^2 + a)^2 - (x^4 + a^2)]Let me expand ((x^2 + a)^2):[x^4 + 2a x^2 + a^2 - x^4 - a^2]Oh, nice! The (x^4) and (a^2) terms cancel out, leaving me with:[2a x^2]So now, the entire expression becomes:[frac{2a x^2}{x (x^2 + a + sqrt{x^4 + a^2})}]Simplifying the denominator, I can factor out an (x):[x (x^2 + a + sqrt{x^4 + a^2})]So, the expression is now:[frac{2a x^2}{x (x^2 + a + sqrt{x^4 + a^2})} = frac{2a x}{x^2 + a + sqrt{x^4 + a^2}}]Hmm, that seems a bit simpler. Maybe I can make a substitution to make this easier. Let me let (y = x^2). Since (x) is positive, (y) will also be positive. Then, the expression becomes:[frac{2a sqrt{y}}{y + a + sqrt{y^2 + a^2}}]Wait, that might not be the most helpful substitution. Let me think again. Maybe instead, I can factor out (x^2) from the terms inside the square root.Looking back at the denominator:[x^2 + a + sqrt{x^4 + a^2}]I can factor out (x^2) from the square root:[sqrt{x^4 + a^2} = x^2 sqrt{1 + frac{a^2}{x^4}}]But that might complicate things further. Alternatively, maybe I can divide both numerator and denominator by (x), to make the expression in terms of (x) and (frac{a}{x}). Let me try that.Dividing numerator and denominator by (x):Numerator: (2a x / x = 2a)Denominator: ((x^2 + a + sqrt{x^4 + a^2}) / x = x + frac{a}{x} + sqrt{x^2 + frac{a^2}{x^2}})So now, the expression is:[frac{2a}{x + frac{a}{x} + sqrt{x^2 + frac{a^2}{x^2}}}]Hmm, this looks a bit more manageable. I notice that both (x + frac{a}{x}) and (x^2 + frac{a^2}{x^2}) are expressions that can be bounded using the AM-GM inequality. Remember, the AM-GM inequality states that for positive real numbers, the arithmetic mean is greater than or equal to the geometric mean.Let me apply AM-GM to (x) and (frac{a}{x}):[x + frac{a}{x} geq 2sqrt{x cdot frac{a}{x}} = 2sqrt{a}]Similarly, for (x^2) and (frac{a^2}{x^2}):[x^2 + frac{a^2}{x^2} geq 2sqrt{x^2 cdot frac{a^2}{x^2}} = 2a]Taking the square root of both sides (since the square root function is increasing):[sqrt{x^2 + frac{a^2}{x^2}} geq sqrt{2a}]So, putting it all together, the denominator:[x + frac{a}{x} + sqrt{x^2 + frac{a^2}{x^2}} geq 2sqrt{a} + sqrt{2a}]Therefore, the entire expression:[frac{2a}{x + frac{a}{x} + sqrt{x^2 + frac{a^2}{x^2}}} leq frac{2a}{2sqrt{a} + sqrt{2a}}]So, the maximum value of the expression is (frac{2a}{2sqrt{a} + sqrt{2a}}). Now, I should check when equality holds to ensure that this maximum is attainable.Equality in AM-GM occurs when the terms are equal. So, for (x + frac{a}{x} geq 2sqrt{a}), equality holds when (x = frac{a}{x}), which implies (x^2 = a), so (x = sqrt{a}).Similarly, for (x^2 + frac{a^2}{x^2} geq 2a), equality holds when (x^2 = frac{a^2}{x^2}), which also implies (x^2 = a), so (x = sqrt{a}).Therefore, when (x = sqrt{a}), both inequalities become equalities, so the denominator reaches its minimum value, and thus the expression reaches its maximum value.Let me plug (x = sqrt{a}) back into the original expression to verify:[frac{(sqrt{a})^2 + a - sqrt{(sqrt{a})^4 + a^2}}{sqrt{a}} = frac{a + a - sqrt{a^2 + a^2}}{sqrt{a}} = frac{2a - sqrt{2a^2}}{sqrt{a}} = frac{2a - asqrt{2}}{sqrt{a}} = frac{a(2 - sqrt{2})}{sqrt{a}} = sqrt{a}(2 - sqrt{2})]Wait, that doesn't seem to match the earlier expression I got. Let me compute (frac{2a}{2sqrt{a} + sqrt{2a}}):First, factor out (sqrt{a}) from the denominator:[2sqrt{a} + sqrt{2a} = sqrt{a}(2 + sqrt{2})]So, the expression becomes:[frac{2a}{sqrt{a}(2 + sqrt{2})} = frac{2sqrt{a}}{2 + sqrt{2}}]Multiply numerator and denominator by (2 - sqrt{2}) to rationalize the denominator:[frac{2sqrt{a}(2 - sqrt{2})}{(2 + sqrt{2})(2 - sqrt{2})} = frac{2sqrt{a}(2 - sqrt{2})}{4 - 2} = frac{2sqrt{a}(2 - sqrt{2})}{2} = sqrt{a}(2 - sqrt{2})]Ah, okay, so both methods give the same result. So, the maximum value is indeed (sqrt{a}(2 - sqrt{2})). But wait, in my initial calculation, I had (frac{2a}{2sqrt{a} + sqrt{2a}}), which simplifies to (sqrt{a}(2 - sqrt{2})). So, that's consistent.Therefore, the maximum value of the given expression is (sqrt{a}(2 - sqrt{2})), achieved when (x = sqrt{a}).I think that makes sense. Let me just recap the steps to make sure I didn't skip anything important.1. I started with the expression (frac{x^2 + a - sqrt{x^4 + a^2}}{x}).2. I multiplied numerator and denominator by the conjugate to rationalize the numerator.3. Simplified the resulting expression to (frac{2a}{x + frac{a}{x} + sqrt{x^2 + frac{a^2}{x^2}}}).4. Applied the AM-GM inequality to the terms in the denominator to find a lower bound.5. Found that the maximum value occurs when (x = sqrt{a}), and verified by plugging it back into the original expression.Everything seems to check out. I don't think I made any mistakes in the algebra or the inequalities. So, I'm confident that the maximum value is indeed (sqrt{a}(2 - sqrt{2})).**Final Answer**The maximum possible value is (boxed{2a left(frac{1}{2sqrt{a} + sqrt{2a}}right)}).Wait, hold on. In my final answer above, I simplified it to (sqrt{a}(2 - sqrt{2})), but the initial expression I had was (frac{2a}{2sqrt{a} + sqrt{2a}}). Let me make sure which form is more appropriate.The problem asks for the maximum value, and both forms are correct, but perhaps the first form is more simplified. Alternatively, maybe expressing it as (sqrt{a}(2 - sqrt{2})) is better because it's a single term.Wait, let me compute both expressions:1. (sqrt{a}(2 - sqrt{2}))2. (frac{2a}{2sqrt{a} + sqrt{2a}})Let me compute the second expression:[frac{2a}{2sqrt{a} + sqrt{2a}} = frac{2a}{sqrt{a}(2 + sqrt{2})} = frac{2sqrt{a}}{2 + sqrt{2}} = sqrt{a} cdot frac{2}{2 + sqrt{2}} = sqrt{a} cdot frac{2(2 - sqrt{2})}{(2 + sqrt{2})(2 - sqrt{2})} = sqrt{a} cdot frac{2(2 - sqrt{2})}{4 - 2} = sqrt{a} cdot frac{2(2 - sqrt{2})}{2} = sqrt{a}(2 - sqrt{2})]So, both expressions are equivalent. Therefore, either form is acceptable, but perhaps the problem expects the answer in the form given in the initial steps, which is (frac{2a}{2sqrt{a} + sqrt{2a}}). Alternatively, it might prefer the simplified radical form (sqrt{a}(2 - sqrt{2})).Looking back at the problem statement, it just asks for the maximum possible value, so either form is correct. However, in the initial solution, the assistant wrote the answer as (boxed{2a left(frac{1}{2sqrt{a} + sqrt{2a}}right)}), which is equivalent to (frac{2a}{2sqrt{a} + sqrt{2a}}).But in my own calculation, I simplified it further to (sqrt{a}(2 - sqrt{2})), which is a more compact form. Maybe I should present it that way.Wait, let me compute (sqrt{a}(2 - sqrt{2})) numerically for a specific value of (a) to see if it matches the original expression when (x = sqrt{a}).Let me take (a = 1). Then, (x = 1).Original expression:[frac{1 + 1 - sqrt{1 + 1}}{1} = frac{2 - sqrt{2}}{1} = 2 - sqrt{2} approx 0.5858]Expression (sqrt{a}(2 - sqrt{2})) when (a = 1):[1 times (2 - sqrt{2}) = 2 - sqrt{2} approx 0.5858]Expression (frac{2a}{2sqrt{a} + sqrt{2a}}) when (a = 1):[frac{2 times 1}{2 times 1 + sqrt{2 times 1}} = frac{2}{2 + sqrt{2}} approx frac{2}{3.4142} approx 0.5858]So, both forms give the same numerical value. Therefore, either form is correct. However, since the problem asks for the maximum possible value, and the expression (sqrt{a}(2 - sqrt{2})) is a simplified form, perhaps that is preferable.But in the initial assistant's solution, they left it as (frac{2a}{2sqrt{a} + sqrt{2a}}). Maybe I should stick with that form unless instructed otherwise.Alternatively, to make it even more simplified, I can factor out (sqrt{a}) from the denominator:[frac{2a}{2sqrt{a} + sqrt{2a}} = frac{2a}{sqrt{a}(2 + sqrt{2})} = frac{2sqrt{a}}{2 + sqrt{2}} = sqrt{a} cdot frac{2}{2 + sqrt{2}} = sqrt{a} cdot frac{2(2 - sqrt{2})}{(2 + sqrt{2})(2 - sqrt{2})} = sqrt{a} cdot frac{2(2 - sqrt{2})}{4 - 2} = sqrt{a}(2 - sqrt{2})]So, both forms are equivalent, but perhaps the problem expects the answer in the form without a fraction in the denominator, which would be (sqrt{a}(2 - sqrt{2})).However, looking back, the assistant's final answer was (boxed{2a left(frac{1}{2sqrt{a} + sqrt{2a}}right)}), which is the same as (frac{2a}{2sqrt{a} + sqrt{2a}}). So, maybe that's the expected form.But in my own calculation, I found that it simplifies to (sqrt{a}(2 - sqrt{2})), which is a more elegant form. I think both are correct, but perhaps the problem expects the answer in the form without the fraction, so I'll go with (sqrt{a}(2 - sqrt{2})).Wait, but in the initial problem statement, the expression is given as (frac{x^2 + a - sqrt{x^4 + a^2}}{x}). When I plug (x = sqrt{a}), I get (sqrt{a}(2 - sqrt{2})), which is a clean answer. So, I think that's the better form.Therefore, I will present the final answer as (boxed{sqrt{a}(2 - sqrt{2})}).But wait, looking back at the assistant's solution, they wrote:"Thus, the maximum value of the given expression is:[frac{2a}{2sqrt{a} + sqrt{2a}} = boxed{2a left(frac{1}{2sqrt{a}+sqrt{2a}}right)},]with equality when (x = sqrt{a})."So, they presented it as (frac{2a}{2sqrt{a} + sqrt{2a}}), which is equivalent to (sqrt{a}(2 - sqrt{2})). So, perhaps either form is acceptable, but since the problem didn't specify, I think both are correct. However, in mathematical problems, often the simplest radical form is preferred, which would be (sqrt{a}(2 - sqrt{2})).But to be consistent with the assistant's solution, maybe I should present it as (frac{2a}{2sqrt{a} + sqrt{2a}}). Alternatively, factor out (sqrt{a}) from the denominator:[frac{2a}{sqrt{a}(2 + sqrt{2})} = frac{2sqrt{a}}{2 + sqrt{2}} = sqrt{a} cdot frac{2}{2 + sqrt{2}} = sqrt{a}(2 - sqrt{2})]So, both forms are correct, but perhaps the problem expects the answer in the form without a fraction in the denominator, which would be (sqrt{a}(2 - sqrt{2})).Alternatively, the problem might prefer the answer in terms of (a) without square roots in the denominator, which would be (frac{2a}{2sqrt{a} + sqrt{2a}}). Hmm.Wait, let me compute both forms for (a = 4):1. (sqrt{4}(2 - sqrt{2}) = 2(2 - sqrt{2}) = 4 - 2sqrt{2} approx 4 - 2.828 = 1.172)2. (frac{2 times 4}{2sqrt{4} + sqrt{2 times 4}} = frac{8}{4 + sqrt{8}} = frac{8}{4 + 2sqrt{2}} = frac{8}{4 + 2.828} approx frac{8}{6.828} approx 1.172)So, both forms give the same numerical value. Therefore, either form is correct. However, since the problem asks for the maximum possible value, and both forms are equivalent, I think either is acceptable. But perhaps the problem expects the answer in the form derived from the AM-GM inequality, which is (frac{2a}{2sqrt{a} + sqrt{2a}}).Alternatively, since (sqrt{a}(2 - sqrt{2})) is a simplified form, it might be preferable. I think I'll go with that.**Final Answer**The maximum possible value is (boxed{sqrt{a}(2 - sqrt{2})}).