Quadratic Equation

1.Let f(x)=4x^3+kx^2-243, where k is a constant. it is given that x+3 is a factor of f(x).

(i)find the value of k
f(-3) = 0
4(-3)^3 + k(-3)^2 - 243 = 0
9k = 351
k = 39
(ii)factorize f(x)
Hence f(x) = 4x^3 + 39x^2 - 243
By long divison,
f(x) = 4x^3 + 39x^2 - 243
= (x+3)(4x^2+27x-81)
= (x+3)(4x-9)(x+9)

2.Let$C be the cost of making a cubical handicraft with a side of length x cm. it is given that C is the sum of two parts, one part varies as x^3 and the other part varies as x^2.when x=5.5,C=7381 and when x=6,C=9072

(i)express C in term of x
C = ax^3 + bx^2 , where a and b are non-zero constants
Put x = 5.5 and C = 7381,
7381 = a(5.5)^3 + b(5.5)^2
7381 = 166.375a + 30.25b
59048 = 1331a + 242b ---(1)
Put x = 6 and C = 9072,
9072 = 216a + 36b ---(2)
(1)*18, 1062864 = 23958a + 4356b ---(3)
(2)*121, 1097712 = 26136a + 4356b ---(4)
(4)-(3), 2178a = 34848
a = 16
Put a = 16 into (2),
59048 = 1331(16) + 242b
b = 156
Hence C = 16x^3 + 156x^2

(ii)If the cost of making a cubical handicraft is$972, find the length of a side of the handicraft.
Put C = 972,
972 = 16x^3 + 156x^2
243 = 4x^3 + 39x^2
4x^3 + 39x^2 - 243 = 0
(x+3)(4x-9)(x+9) = 0 (From Q1)
x = -3(rejected) or x = 9/4 or x = -9(rejected)
Hence the length of a side of the handicraft is 9/4 m.