answer:1. Let ( x_0 ) be the unique root of the quadratic polynomial ( g(x) ). 2. This implies ( g(x_0) = 0 ) and ( g(x) ) does not change its sign for ( x neq x_0 ). 3. Consider the polynomial ( f(x) = g(ax + b) + g(cx + d) ). Given that ( f(x) ) has exactly one root, let ( x_1 ) be this unique root. 4. The root ( x_1 ) makes both ( g(ax + b) ) and ( g(cx + d) ) zero simultaneously, which suggests: [ ax_1 + b = x_0 quad text{and} quad cx_1 + d = x_0. ] 5. Solving these equations simultaneously, we set: [ ax_1 + b = cx_1 + d, ] leading to: [ ax_1 - cx_1 = d - b, ] and therefore: [ (a - c)x_1 = d - b. ] 6. Since ( a neq c ) (by the problem's condition), we can solve for ( x_1 ): [ x_1 = frac{d - b}{a - c}. ] 7. To find ( x_0 ), which is the root we are asked to find, substitute ( x_1 ) into either ( ax_1 + b = x_0 ) or ( cx_1 + d = x_0 ). Using, for example, ( ax_1 + b = x_0 ): [ x_0 = aleft(frac{d - b}{a - c}right) + b. ] 8. Simplifying the expression for ( x_0 ): [ x_0 = frac{a(d - b) + b(a - c)}{a - c} = frac{ad - ab + ab - bc}{a - c} = frac{ad - bc}{a - c}. ] 9. Thus, the unique root ( x_0 ) of the polynomial ( g(x) ) is given by: [ boxed{ frac{ad - bc}{a - c} }. ]

answer:1. **Square Details:** - Area of the square = 256 - Side length of square = sqrt{256} = 16 units - Perimeter of square (circumference of circle) = 4 times 16 = 64 units 2. **Rectangle Details:** - Assuming rectangle width is w and height is h with w = 2h. - Rectangle's perimeter = 2w + 2h = 64 - Substituting w = 2h into the perimeter equation, 2(2h) + 2h = 64 implies 6h = 64 implies h = frac{64}{6} approx 10.67 - Therefore, width w = 2 times 10.67 approx 21.33 3. **Calculating the Circle's Radius and Area:** - Circle's radius r from its circumference = frac{Circumference}{2pi} = frac{64}{2pi} = frac{32}{pi} - Circle's area = pi r^2 = pi left(frac{32}{pi}right)^2 = frac{1024}{pi} boxed{frac{1024}{pi}}

answer:Let's assume the power function is y=f(x)=x^{alpha}. Since it passes through the point (-2, -frac{1}{8}), we have: -frac{1}{8}=(-2)^{alpha} Solving this equation, we get alpha=-3. Therefore, f(x)=x^{-3}. To find the value of x that makes f(x)=27, we set: f(x)=27=x^{-3} Solving this equation, we find x=frac{1}{3}. Hence, the answer is: boxed{frac{1}{3}}. First, we set up the equation of the power function and substitute the point (-2, -frac{1}{8}) to find the value of alpha. Then, we substitute 27 into the equation to find the value of x. This problem tests the method of finding the equation of a power function, which involves using the method of undetermined coefficients. It is a basic question.

answer:1. **Determine the amount of rice per bag**: The total amount of rice is ( 11410 , text{kg} ) and it is distributed equally into 3260 bags. Therefore, to find the amount of rice in each bag, we perform the division: [ frac{11410 , text{kg}}{3260 , text{bags}} = 3.5 , text{kg per bag} ] 2. **Calculate the number of days to use one bag**: Since the family uses ( 0.25 , text{kg} ) of rice each day, we need to determine how many days it would take to completely use one bag of rice that contains ( 3.5 , text{kg} ). This can be calculated as: [ frac{3.5 , text{kg}}{0.25 , text{kg/day}} = 14 , text{days} ] 3. **Verify the calculation**: Considering the calculation: [ frac{3.5}{0.25} = frac{3.5 times 4}{0.25 times 4} = frac{14}{1} = 14 ] This confirms the previous result. # Conclusion: It would take the family 14 days to use up one bag of rice. [ boxed{D} ]