answer：Alright, so I have these three problems to solve. Let me take them one by one. I remember the concept of "whole thinking" from the introduction. It's about treating a part of an expression as a single unit and combining like terms. That should help simplify things, right?Starting with problem (1): I need to combine (4(a - b)^2 - 6(a - b)^2 + 8(a - b)^2). Okay, so each term here has ((a - b)^2), which is the whole. So, similar to the examples given, I can factor that out and just combine the coefficients.Let me write that down:(4(a - b)^2 - 6(a - b)^2 + 8(a - b)^2)This is like (4x - 6x + 8x) where (x = (a - b)^2). So, combining the coefficients:(4 - 6 + 8 = (4 + 8) - 6 = 12 - 6 = 6). So, the result should be (6(a - b)^2). That seems straightforward.Moving on to problem (2): Given (x^2 - 2y = 1), find the value of (2021x^2 - 4042y + 1). Hmm, okay. So, I need to express the given expression in terms of the equation provided.Looking at the expression (2021x^2 - 4042y + 1), I notice that 2021 and 4042 are multiples. Let me see, 4042 divided by 2021 is 2. So, 4042y is 2 times 2021y. Wait, but in the given equation, we have (x^2 - 2y = 1). Maybe I can factor out 2021 from the first two terms.Let me try that:(2021x^2 - 4042y + 1 = 2021(x^2 - 2y) + 1)Yes, that works because 2021 times (x^2) is (2021x^2), and 2021 times (-2y) is (-4042y). So, now, from the given equation, (x^2 - 2y = 1), so substituting that in:(2021(1) + 1 = 2021 + 1 = 2022). So, the value is 2022. That makes sense.Now, problem (3): Given (a - 2b = 2), (2b - c = -5), and (c - d = 9), find the value of ((a - c) + (2b - d) - (2b - c)). Hmm, okay. Let me parse this step by step.First, let me write down what I need to find:((a - c) + (2b - d) - (2b - c))Let me simplify this expression before plugging in any values. Maybe I can combine like terms.Expanding the expression:(a - c + 2b - d - 2b + c)Wait, let's see:- The first term is (a - c).- The second term is (2b - d).- The third term is subtracted, so it's (-2b + c).So, combining all these:(a - c + 2b - d - 2b + c)Now, let's combine like terms:- (a) remains as is.- (-c + c = 0).- (2b - 2b = 0).- (-d) remains.So, the entire expression simplifies to (a - d). Interesting, so the value we need is (a - d).Now, how do I find (a - d) given the equations?Given:1. (a - 2b = 2)2. (2b - c = -5)3. (c - d = 9)I need to express (a - d) in terms of these equations. Let me see if I can link these equations together.From equation 1: (a = 2b + 2).From equation 2: (2b - c = -5), so (c = 2b + 5).From equation 3: (c - d = 9), so (d = c - 9). But since (c = 2b + 5), substituting that in, (d = (2b + 5) - 9 = 2b - 4).So now, I have expressions for (a) and (d) in terms of (b):- (a = 2b + 2)- (d = 2b - 4)Therefore, (a - d = (2b + 2) - (2b - 4))Let me compute that:(2b + 2 - 2b + 4 = (2b - 2b) + (2 + 4) = 0 + 6 = 6)So, (a - d = 6). Therefore, the value of the original expression is 6.Let me double-check my steps to make sure I didn't make any mistakes.Starting with the expression ((a - c) + (2b - d) - (2b - c)), I expanded it correctly:(a - c + 2b - d - 2b + c), which simplifies to (a - d). That seems right.Then, using the given equations:1. (a = 2b + 2)2. (c = 2b + 5)3. (d = c - 9 = 2b + 5 - 9 = 2b - 4)Thus, (a - d = (2b + 2) - (2b - 4) = 6). Yep, that checks out.So, all three problems solved. Let me recap:1. Combined the coefficients of ((a - b)^2) to get (6(a - b)^2).2. Factored out 2021 from the expression and substituted the given equation to get 2022.3. Simplified the expression to (a - d), then expressed (a) and (d) in terms of (b) and found the difference to be 6.Feeling confident about these answers.**Final Answer**(1) boxed{6(a - b)^2}(2) boxed{2022}(3) boxed{6}

answer：Okay, so I have this math problem here. It says that the square root of the real number ( a + 3 ) is ( pm 4 ). I need to find the arithmetic square root and the cubic root of the real number ( 5a - 1 ). Hmm, let me try to figure this out step by step.First, the problem mentions that the square root of ( a + 3 ) is ( pm 4 ). Wait, square roots can be both positive and negative, right? So if the square root is ( pm 4 ), that means ( a + 3 ) must be equal to ( 4^2 ) or ( (-4)^2 ). But both of those are 16, right? So ( a + 3 = 16 ).Let me write that down:( sqrt{a + 3} = pm 4 )Squaring both sides, we get:( a + 3 = (pm 4)^2 )Which simplifies to:( a + 3 = 16 )So, solving for ( a ), subtract 3 from both sides:( a = 16 - 3 )( a = 13 )Okay, so ( a ) is 13. Now, the problem asks for the arithmetic square root and the cubic root of ( 5a - 1 ). Let me compute ( 5a - 1 ) first.Substituting ( a = 13 ):( 5a - 1 = 5 times 13 - 1 )Calculating that:( 5 times 13 = 65 )So,( 65 - 1 = 64 )Therefore, ( 5a - 1 = 64 ).Now, I need to find the arithmetic square root and the cubic root of 64.Starting with the arithmetic square root. The arithmetic square root is the non-negative root, so it's just the positive square root. So, what number squared gives 64?Well, 8 squared is 64 because ( 8 times 8 = 64 ). So, the arithmetic square root of 64 is 8.Next, the cubic root. The cubic root of a number is a value that, when multiplied by itself three times, gives the original number. So, what number cubed is 64?Let me think. 4 cubed is 64 because ( 4 times 4 times 4 = 64 ). So, the cubic root of 64 is 4.Wait, just to make sure I didn't make a mistake. Let me double-check my calculations.First, ( a + 3 = 16 ) leads to ( a = 13 ). Then, ( 5a - 1 = 5 times 13 - 1 = 65 - 1 = 64 ). That seems correct.Square root of 64 is 8 because ( 8^2 = 64 ). Cubic root of 64 is 4 because ( 4^3 = 64 ). Yep, that all adds up.So, I think I've got it. The arithmetic square root is 8, and the cubic root is 4.**Final Answer**The arithmetic square root of (5a - 1) is boxed{8} and the cubic root is boxed{4}.

answer：Okay, so I have this problem about a parabola and a line intersecting it. Let me try to figure it out step by step. The parabola is given by the equation ( y^2 = 4x ). I remember that for a parabola in the form ( y^2 = 4ax ), the focus is at ( (a, 0) ). Comparing that to our equation, ( 4a = 4 ), so ( a = 1 ). That means the focus ( F ) is at ( (1, 0) ).Now, the problem says a line passes through ( F ) with an inclination angle of ( frac{pi}{4} ). I think the inclination angle is the angle the line makes with the positive x-axis. So, if the angle is ( frac{pi}{4} ), that's 45 degrees. The slope of such a line should be ( tan(frac{pi}{4}) ), which is 1. So, the slope ( m ) is 1.Since the line passes through the focus ( F(1, 0) ) and has a slope of 1, I can write its equation using the point-slope form. The point-slope formula is ( y - y_1 = m(x - x_1) ). Plugging in the values, we get ( y - 0 = 1 cdot (x - 1) ), which simplifies to ( y = x - 1 ).So, the equation of the line is ( y = x - 1 ). Now, this line intersects the parabola ( y^2 = 4x ) at points ( A ) and ( B ). I need to find these points of intersection to determine the distance between ( A ) and ( B ).To find the points of intersection, I can substitute ( y = x - 1 ) into the equation of the parabola. Let's do that:Substitute ( y ) into ( y^2 = 4x ):( (x - 1)^2 = 4x )Let me expand the left side:( x^2 - 2x + 1 = 4x )Now, bring all terms to one side to form a quadratic equation:( x^2 - 2x + 1 - 4x = 0 )Simplify the like terms:( x^2 - 6x + 1 = 0 )So, the quadratic equation is ( x^2 - 6x + 1 = 0 ). I can solve this using the quadratic formula. The quadratic formula is ( x = frac{-b pm sqrt{b^2 - 4ac}}{2a} ), where ( a = 1 ), ( b = -6 ), and ( c = 1 ).Plugging in the values:( x = frac{-(-6) pm sqrt{(-6)^2 - 4 cdot 1 cdot 1}}{2 cdot 1} )Simplify:( x = frac{6 pm sqrt{36 - 4}}{2} )( x = frac{6 pm sqrt{32}}{2} )( sqrt{32} ) can be simplified to ( 4sqrt{2} ), so:( x = frac{6 pm 4sqrt{2}}{2} )Divide numerator terms by 2:( x = 3 pm 2sqrt{2} )So, the x-coordinates of points ( A ) and ( B ) are ( 3 + 2sqrt{2} ) and ( 3 - 2sqrt{2} ). Now, let's find the corresponding y-coordinates using the equation of the line ( y = x - 1 ).For ( x = 3 + 2sqrt{2} ):( y = (3 + 2sqrt{2}) - 1 = 2 + 2sqrt{2} )So, point ( A ) is ( (3 + 2sqrt{2}, 2 + 2sqrt{2}) ).For ( x = 3 - 2sqrt{2} ):( y = (3 - 2sqrt{2}) - 1 = 2 - 2sqrt{2} )So, point ( B ) is ( (3 - 2sqrt{2}, 2 - 2sqrt{2}) ).Now, I need to find the distance between points ( A ) and ( B ). The distance formula between two points ( (x_1, y_1) ) and ( (x_2, y_2) ) is ( sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ).Let me compute the differences first:( x_2 - x_1 = (3 - 2sqrt{2}) - (3 + 2sqrt{2}) = -4sqrt{2} )( y_2 - y_1 = (2 - 2sqrt{2}) - (2 + 2sqrt{2}) = -4sqrt{2} )Now, square these differences:( (-4sqrt{2})^2 = 16 cdot 2 = 32 )So, both ( (x_2 - x_1)^2 ) and ( (y_2 - y_1)^2 ) are 32.Add them together:( 32 + 32 = 64 )Take the square root of 64:( sqrt{64} = 8 )So, the distance between points ( A ) and ( B ) is 8 units.Wait, let me double-check my calculations to make sure I didn't make any mistakes. The quadratic equation gave me x-values of ( 3 pm 2sqrt{2} ), which seems correct. Plugging back into the line equation, the y-values are ( 2 pm 2sqrt{2} ). Then, the differences in x and y are both ( -4sqrt{2} ), which when squared give 32 each. Adding them gives 64, and the square root of 64 is indeed 8. So, I think that's correct.Alternatively, I remember that for a parabola ( y^2 = 4ax ), the length of the focal chord (a chord passing through the focus) with a slope ( m ) can be found using the formula ( frac{4a(1 + m^2)}{m^2} ). Wait, is that right? Let me think.Actually, I think the formula for the length of the focal chord is ( frac{4a}{sin^2 theta} ), where ( theta ) is the angle the chord makes with the x-axis. Since here the angle is ( frac{pi}{4} ), ( sin frac{pi}{4} = frac{sqrt{2}}{2} ), so ( sin^2 frac{pi}{4} = frac{1}{2} ). Then, the length would be ( frac{4a}{1/2} = 8a ). Since ( a = 1 ), the length is 8. That matches my earlier result. So, that's a good confirmation.Therefore, I can be confident that the distance ( |AB| ) is 8.**Final Answer**The length of segment ( AB ) is boxed{8}.

answer：First, I'll define variables to represent the number of candies each person has. Let ( C ) be the number of candies Corey has. Since Tapanga has 8 more candies than Corey, Tapanga has ( C + 8 ) candies.Next, I'll set up an equation based on the total number of candies they have together. The sum of Corey's and Tapanga's candies is 66, so:[C + (C + 8) = 66]Simplifying the equation:[2C + 8 = 66]Subtracting 8 from both sides:[2C = 58]Finally, dividing both sides by 2:[C = 29]Therefore, Corey has 29 candies.

answer：Okay, so I have this problem here where I need to find the minimum value of the sum of two distances, d₁ and d₂, from a moving point P on the parabola y² = 4x to two given lines, l₁ and l₂. The lines are l₁: 4x - 3y + 16 = 0 and l₂: x = -1. First, I should recall the formula for the distance from a point to a line. The distance from a point (x₀, y₀) to the line ax + by + c = 0 is given by |ax₀ + by₀ + c| / sqrt(a² + b²). So, I can use this formula to express both d₁ and d₂ in terms of the coordinates of point P.Since P is on the parabola y² = 4x, I can parametrize P using the standard parametric equations for a parabola. For y² = 4x, the parametric equations are x = at² and y = 2at, where a is the focal length. Comparing with y² = 4x, we have 4a = 4, so a = 1. Therefore, the parametric coordinates of P can be written as (t², 2t). This will make it easier to express d₁ and d₂ in terms of a single variable t.So, let me write down the coordinates of P as (t², 2t). Now, let's compute d₁, the distance from P to l₁: 4x - 3y + 16 = 0.Using the distance formula:d₁ = |4*t² - 3*(2t) + 16| / sqrt(4² + (-3)²) = |4t² - 6t + 16| / 5.Similarly, d₂ is the distance from P to l₂: x = -1. Since l₂ is a vertical line, the distance from P = (t², 2t) to l₂ is simply the horizontal distance between x = t² and x = -1. So, d₂ = |t² - (-1)| = |t² + 1|. Since t² is always non-negative, t² + 1 is always positive, so we can drop the absolute value: d₂ = t² + 1.Therefore, the total distance we need to minimize is d₁ + d₂ = (|4t² - 6t + 16| / 5) + (t² + 1). Wait a second, the numerator in d₁ is 4t² - 6t + 16. Let me check if this quadratic ever becomes negative. The quadratic 4t² - 6t + 16 has a discriminant D = (-6)² - 4*4*16 = 36 - 256 = -220, which is negative. That means the quadratic never crosses zero and is always positive because the coefficient of t² is positive. So, |4t² - 6t + 16| is just 4t² - 6t + 16. Therefore, d₁ simplifies to (4t² - 6t + 16)/5.So, now, the expression to minimize is:d₁ + d₂ = (4t² - 6t + 16)/5 + t² + 1.Let me combine these terms. First, let's write t² as 5t²/5 to have a common denominator:(4t² - 6t + 16)/5 + (5t²)/5 + 1.Combine the terms:(4t² - 6t + 16 + 5t²)/5 + 1 = (9t² - 6t + 16)/5 + 1.Now, 1 can be written as 5/5, so:(9t² - 6t + 16 + 5)/5 = (9t² - 6t + 21)/5.Therefore, the expression simplifies to (9t² - 6t + 21)/5. So, we need to find the minimum value of this quadratic function in terms of t.Since this is a quadratic function in t, and the coefficient of t² is positive (9/5), the function opens upwards, so its minimum occurs at the vertex.The vertex of a quadratic function at² + bt + c is at t = -b/(2a). Here, a = 9/5, b = -6/5.Wait, hold on. Let me write the function as f(t) = (9t² - 6t + 21)/5. So, f(t) = (9/5)t² - (6/5)t + 21/5.So, in standard form, a = 9/5, b = -6/5, c = 21/5.Therefore, the vertex is at t = -b/(2a) = -(-6/5)/(2*(9/5)) = (6/5)/(18/5) = (6/5)*(5/18) = 6/18 = 1/3.So, the minimum occurs at t = 1/3.Now, let's compute f(1/3):f(1/3) = (9*(1/3)^2 - 6*(1/3) + 21)/5.Compute each term:9*(1/3)^2 = 9*(1/9) = 1.-6*(1/3) = -2.So, 1 - 2 + 21 = 20.Therefore, f(1/3) = 20/5 = 4.So, the minimum value of d₁ + d₂ is 4.Wait, hold on. Let me double-check my calculations because 9*(1/3)^2 is 1, -6*(1/3) is -2, and 21 is 21. So, 1 - 2 + 21 is indeed 20, and 20/5 is 4. So, that seems correct.But just to be thorough, let me verify if I correctly transformed the expression.We had d₁ = (4t² - 6t + 16)/5 and d₂ = t² + 1.So, d₁ + d₂ = (4t² - 6t + 16)/5 + t² + 1.Expressed as:(4t² - 6t + 16)/5 + (5t²)/5 + 5/5 = (4t² - 6t + 16 + 5t² + 5)/5 = (9t² - 6t + 21)/5.Yes, that's correct.So, the quadratic is (9t² - 6t + 21)/5, which we found the minimum at t = 1/3, giving 4.Therefore, the minimum value of d₁ + d₂ is 4.But wait, let me think again. Is there another way to approach this problem? Maybe using reflection properties of parabolas?I remember that for a parabola, the definition is the set of points equidistant from the focus and the directrix. The given parabola is y² = 4x, which has its focus at (1, 0) and directrix at x = -1, which is exactly the line l₂: x = -1.So, in this case, d₂ is the distance from P to the directrix, which is equal to the distance from P to the focus (1, 0). So, d₂ = distance from P to (1, 0).Therefore, d₁ + d₂ is equal to d₁ + distance from P to (1, 0). So, we can rephrase the problem as finding the minimum of d₁ + distance from P to (1, 0), where P is on the parabola y² = 4x.But d₁ is the distance from P to the line l₁: 4x - 3y + 16 = 0. So, we need to minimize the sum of the distance from P to l₁ and the distance from P to the focus (1, 0).Hmm, is there a geometric interpretation here? Maybe using reflection properties or something?Alternatively, perhaps using calculus, as I did before, is the straightforward way. But since I got an answer of 4, I should verify if that's correct.Wait, let me compute d₁ + d₂ at t = 1/3.So, P is (t², 2t) = ((1/3)^2, 2*(1/3)) = (1/9, 2/3).Compute d₁: distance from (1/9, 2/3) to l₁: 4x - 3y + 16 = 0.Using the distance formula:|4*(1/9) - 3*(2/3) + 16| / 5 = |4/9 - 2 + 16| / 5.Compute numerator:4/9 - 2 + 16 = 4/9 + 14 = (4 + 126)/9 = 130/9.So, |130/9| /5 = (130/9)/5 = 26/9 ≈ 2.888...Compute d₂: distance from (1/9, 2/3) to x = -1 is |1/9 - (-1)| = |1/9 + 1| = 10/9 ≈ 1.111...So, d₁ + d₂ = 26/9 + 10/9 = 36/9 = 4. So, that's correct.Therefore, the minimum value is indeed 4.Alternatively, if I think about reflecting the focus over the line l₁ and then finding the distance from the reflection to the directrix or something, but that might complicate things.But since the calculus approach worked out and the verification shows that at t = 1/3, the sum is 4, which is the minimum, I think that's solid.Therefore, the minimum value of d₁ + d₂ is 4.**Final Answer**The minimum value of (d_1 + d_2) is boxed{4}.

answer：First, I need to determine the initial amounts of each component in the 30-liter paint mixture. The mixture is 25% red tint, 30% yellow tint, and 45% water.Calculating the amount of red tint:25% of 30 liters is 0.25 * 30 = 7.5 liters.Calculating the amount of yellow tint:30% of 30 liters is 0.30 * 30 = 9 liters.Calculating the amount of water:45% of 30 liters is 0.45 * 30 = 13.5 liters.Next, 5 liters of yellow tint are added to the original mixture. This increases the total amount of yellow tint to:9 liters + 5 liters = 14 liters.The total volume of the new mixture becomes:30 liters + 5 liters = 35 liters.Finally, to find the percentage of yellow tint in the new mixture, I divide the amount of yellow tint by the total volume and multiply by 100:(14 liters / 35 liters) * 100 = 40%.