answer:To solve the problem of finding the discriminant of the quadratic equation (x^2 - 4x + 3 = 0), we follow these steps:1. Identify the coefficients of the quadratic equation: The general form of a quadratic equation is (ax^2 + bx + c = 0). In this case, (a = 1), (b = -4), and (c = 3).2. Use the formula for the discriminant: The discriminant (Delta) of a quadratic equation (ax^2 + bx + c = 0) is given by [ Delta = b^2 - 4ac ]3. Substitute the values of (a), (b), and (c) into the discriminant formula and compute the result.Let's use Python and sympy to compute the discriminant.```pythonimport sympy as sp# Define the coefficientsa = 1b = -4c = 3# Calculate the discriminantdiscriminant = b2 - 4*a*cprint(discriminant)``````output4```The discriminant of the quadratic equation (x^2 - 4x + 3 = 0) is (boxed{4}).

answer:To solve for ( n ) given the constraints (mathop{text{lcm}}[24,n]=72) and (mathop{text{lcm}}[n,27]=108), we can use the property of the least common multiple (LCM). Recall that for any two integers (a) and (b):[mathop{text{lcm}}[a, b] = frac{|a cdot b|}{mathop{text{gcd}}(a, b)}]where (mathop{text{gcd}}(a, b)) is the greatest common divisor of (a) and (b).Let's break down the problem step by step.Given:[mathop{text{lcm}}[24, n] = 72][mathop{text{lcm}}[n, 27] = 108]We first look at the prime factorizations:- (24 = 2^3 times 3)- (72 = 2^3 times 3^2)- (27 = 3^3)- (108 = 2^2 times 3^3)From (mathop{text{lcm}}[24, n] = 72), (n) must have factors that combine with (24)'s factors to produce (72). Thus, (n) must have at least the factor (3^2) and at most (2^2 times 3^2):From (mathop{text{lcm}}[n, 27] = 108), (n) must have factors that combine with (27)'s factors to produce (108). Thus, (n) must have at least the factor (2^2) and at most (2^2 times 3^2). We now check the intersections and restrictions:- ( n = 2^2 times 3^a ) where (a) must be (2).Thus:[n = 2^2 times 3^2 = 4 times 9 = 36]Let's confirm using Python to ensure our reasoning and calculations are correct.```pythonimport mathdef find_n(): for n in range(1, 1000): # Arbitrarily chosen upper limit to search for n if math.lcm(24, n) == 72 and math.lcm(n, 27) == 108: return n return Nonen = find_n()print(n)``````output36```The value of ( n ) that satisfies both (mathop{text{lcm}}[24, n]=72) and (mathop{text{lcm}}[n, 27]=108) is (boxed{36}).

answer:To solve this problem, let's break it down into smaller steps and use Python with sympy for symbolic computation:1. Let the original purchase price be (P).2. Let the selling price be (S).3. The original profit margin is (x%). This means that the profit is (S - P) and the profit margin based on the purchase price is (frac{S - P}{P} times 100 = x%).After reducing the purchase price by 8%, the new purchase price is (0.92P). The new profit margin based on the new purchase price is given to be ((x + 10)%). So, the new profit margin is (frac{S - 0.92P}{0.92P} times 100 = (x + 10)%).We need to solve for (x). Let's set up the equations and use sympy to solve them.First, let's convert our given percentages into decimal form and set up the equations:[frac{S - P}{P} = frac{x}{100}][frac{S - 0.92P}{0.92P} = frac{x + 10}{100}]Now, we can use sympy to solve these equations for (x).```pythonimport sympy as sp# Define the variablesx = sp.symbols('x')P = sp.symbols('P', positive=True)S = sp.symbols('S', positive=True)# Original profit margin equationoriginal_profit_margin = (S - P) / P - x / 100# New profit margin equation after reducing the purchase price by 8%new_profit_margin = (S - 0.92 * P) / (0.92 * P) - (x + 10) / 100# Solve the equationssolution = sp.solve((original_profit_margin, new_profit_margin), (S, x))final_x = solution[x]print(final_x)``````output15.0000000000000```The value of ( x % ) for which the profit margin increases from ( x % ) to ( (x + 10) % ) when the purchase price is reduced by 8% is (boxed{15%}).So the correct option is (B) (15%).

answer:To solve this problem, let's break it down into three parts: Part 1: Prove that ( f(x) ) is a periodic function.Given that ( f(x) ) is an odd function, we have ( f(-x) = -f(x) ). Additionally, the problem states that ( f(x + 2) = -f(x) ). We need to show that there exists a period ( T ) such that ( f(x + T) = f(x) ) for all ( x ).Let's investigate this property using the given information:1. ( f(x + 2) = -f(x) )2. ( f(x + 4) = f((x + 2) + 2) = f(x + 2 + 2) = -f(x + 2) = -(-f(x)) = f(x) )Thus, ( f(x + 4) = f(x) ), proving that ( f(x) ) is periodic with period ( T = 4 ). Part 2: When ( x in [2, 4] ), find the expression for ( f(x) ).We know ( f(x + 2) = -f(x) ). Thus, for ( x in [2, 4] ):- ( f(x) = -f(x - 2) ) for ( x in [2, 4] )- Since ( f(x) = 2x - x^2 ) for ( x in [0, 2] ), we have: - ( f(x - 2) = 2(x - 2) - (x - 2)^2 )Let's compute ( f(x) ) for ( x in [2, 4] ):[f(x) = -f(x - 2) = -left(2(x - 2) - (x - 2)^2right)]Simplify to get the expression:[f(x) = -left(2x - 4 - (x^2 - 4x + 4)right) = -left(2x - 4 - x^2 + 4x - 4right) = -(-x^2 + 6x - 8) = x^2 - 6x + 8] Part 3: Calculate ( f(0) + f(1) + f(2) + ldots + f(2011) ).Since ( f(x) ) is periodic with period 4, we can sum ( f(x) ) over each period and repeat the sum over the full range from 0 to 2011.Let's first find the sum for one period ( [0, 4] ):- ( f(0) = 0 )- ( f(1) = 2 cdot 1 - 1^2 = 1 )- ( f(2) = 2 cdot 2 - 2^2 = 0 )- ( f(3) = 3^2 - 6 cdot 3 + 8 = 9 - 18 + 8 = -1 )The sum for one period is:[f(0) + f(1) + f(2) + f(3) = 0 + 1 + 0 - 1 = 0]Since ( f(x) ) is periodic with period 4, the sum ( f(0) + f(1) + ldots + f(2011) ) will be the sum of 503 periods (since 2012/4 = 503):[503 times 0 = 0]Let's use Python and sympy to verify our results.```pythonimport sympy as sp# Define the function in the range [0, 2]def f_0_2(x): return 2 * x - x 2# Define the function in the range [2, 4]def f_2_4(x): return x 2 - 6 * x + 8# Sum the values of f(x) from 0 to 3 (one period)sum_one_period = f_0_2(0) + f_0_2(1) + f_0_2(2) + f_2_4(3)# Calculate the total sum for 503 periodstotal_sum = sum_one_period * 503print(total_sum)``````output0```As we can see from the output, the sum of ( f(x) ) over one period is indeed ( 0 ), confirming our previous calculations. Let's summarize the results:1. Proof of Periodicity: - ( f(x + 2) = -f(x) ) - ( f(x + 4) = f(x) ) - Therefore, ( f(x) ) is periodic with period ( T = 4 ).2. Expression for ( f(x) ) in ( [2, 4] ): - For ( x in [2, 4] ): [ f(x) = x^2 - 6x + 8 ]3. Sum Calculation: - The sum for one period ( [0, 4] ): [ f(0) + f(1) + f(2) + f(3) = 0 + 1 + 0 - 1 = 0 ] - The total sum ( f(0) + f(1) + ldots + f(2011) ): [ 503 times 0 = 0 ]Thus, the final answer for ( f(0) + f(1) + f(2) + ldots + f(2011) ) is ( boxed{0} ).

answer:To solve the problem and find the derivative of the function ( f(x) = x^2 + 2x ) when ( x = 0 ), we can follow these steps:1. Define the function ( f(x) ).2. Compute its derivative ( f'(x) ).3. Evaluate the derivative at ( x = 0 ).Let's implement this in Python using the `sympy` library:```pythonimport sympy as sp# Define the variablex = sp.symbols('x')# Define the function f(x)f = x2 + 2*x# Compute the derivative of f(x)f_prime = sp.diff(f, x)# Evaluate the derivative at x = 0f_prime_at_0 = f_prime.subs(x, 0)# Print the resultprint(f_prime_at_0)``````output2```The derivative of the function ( f(x) = x^2 + 2x ) at ( x = 0 ) is ( f'(0) = 2 ).Thus, the final answer is (boxed{2}).

answer:To solve the given problem, we need to use properties of triangles and medians. The medians in a triangle have unique properties, and additional geometric insights can help us solve the problem. Here's the reasoning and step-by-step process:1. Set up the problem: We have triangle ( ABC ) with: - Side ( BC = 28 ) - Side ( AC = 44 ) - The medians drawn from vertices ( A ) and ( B ) are perpendicular.We'll denote:- ( AB = c )- ( BC = a = 28 )- ( AC = b = 44 )2. Median properties: The medians of a triangle intersect at the centroid, and the centroid divides each median into a ratio of 2:1.3. Use coordinate geometry: It would be useful to place the triangle in a coordinate system to leverage the perpendicularity of the medians and get the side ( AB ). Steps:1. Place vertex ( A ) at the origin, so ( A = (0,0) ).2. Place vertex ( B ) at ( (c, 0) ). 3. Use the coordinates formula and median properties to find vertex ( C ).4. Utilize the perpendicularity condition of the medians. Median coordinates:- Median from ( A ) meets at point ( D ), midpoint of ( BC ).- Median from ( B ) meets at point ( E ), midpoint of ( AC ).Let's employ coordinate geometry and Python for precise solution checking:1. Coordinates of points of the medians midpoints (D) and (E):- Coordinates of (C): Let's assume (C) is at ((x, y)). Using Pythagorean theorem in triangles:- ( AC = sqrt{x^2 + y^2} = 44 )- ( BC = sqrt{(x-c)^2 + y^2} = 28 )2. Calculate (D) and (E):- (D = left(frac{x+c}{2}, frac{y}{2}right))- (E = left(frac{x}{2}, frac{y}{2}right))3. Use the condition that medians from (A) and (B) are perpendicular:To solve it, let's setup equations and solve for ( c ).Given (a) and (b), the triangles' property usage and midpoint with perpendicular conditions can be numerically solved using equations backup by Python.We can code this approach to solve for (c).```pythonfrom sympy import symbols, Eq, solve, sqrt# Given Valuesa = 28 # BCb = 44 # AC# Variablesc = symbols('c') # ABx, y = symbols('x y') # Coordinates of point C# Define the equations based on distances involving AC and BCeq1 = Eq(sqrt(x2 + y2), b) # Distance ACeq2 = Eq(sqrt((x - c)2 + y2), a) # Distance BC# Solving for x and y in terms of c using givensolutions = solve((eq1, eq2), (x, y))# The solutions will be in terms of cprint(solutions)``````output[((c2 + 1152)/(2*c), -sqrt(-(c - 72)*(c - 16)*(c + 16)*(c + 72))/(2*c)), ((c2 + 1152)/(2*c), sqrt(-(c - 72)*(c - 16)*(c + 16)*(c + 72))/(2*c))]```The solutions for ( C ) in terms of ( c ) are:[ C = left( frac{c^2 + 1152}{2c}, pm frac{sqrt{-(c - 72)(c - 16)(c + 16)(c + 72)}}{2c} right) ]Now we need to ensure that the medians from ( A ) and ( B ) are perpendicular. This involves:1. Calculate the midpoints (D) and (E): - ( D = left( frac{x+c}{2}, frac{y}{2} right) ) - ( E = left( frac{x}{2}, frac{y}{2} right) ) 2. Confirm perpendicularity condition (dot product of directional vectors being zero).Perpendicularity of vectors:[ left( frac{partial D}{partial x} - 0, frac{partial D}{partial y} - 0 right) cdot left( frac{partial E}{partial x} - c, frac{partial E}{partial y} - 0 right) = 0 ]We'll examine this step algebraically using Python.```python# Coordinates based on our solutionx_exp, y_exp = (c2 + 1152) / (2 * c), sqrt(-(c - 72)*(c - 16)*(c + 16)*(c + 72)) / (2 * c)# Midpoints D and EDx, Dy = (x_exp + c) / 2, y_exp / 2Ex, Ey = x_exp / 2, y_exp / 2# Vectors for medians AD and BEADx, ADy = Dx, DyBEx, BEy = Ex - c, Ey# Dot product for perpendicularity conditiondot_product = ADx * BEx + ADy * BEy# Solve for cc_solution = solve(Eq(dot_product, 0), c)print(c_solution)``````output[-4*sqrt(34), 4*sqrt(34)]```The possible solutions for ( c ) (the length of side ( AB )) are:[ c = -4sqrt{34} ][ c = 4sqrt{34} ]Since the length of a side cannot be negative, we discard ( -4sqrt{34} ). Hence, the valid side length ( AB ) is:[ c = 4sqrt{34} ] Area of the square with side ( AB )The area of the square is given by the square of the side length ( AB ):[text{Area} = c^2 = (4sqrt{34})^2]Let's calculate the numerical value.```python# Calculate the area of the squarec_value = 4 * sqrt(34)area_square = c_value2print(area_square.evalf())``````output544.000000000000```The area of the square with side ( AB ) is:[boxed{544}]