A candy store sells 2 ounces of chocolate for $0.80 and 3 ounces of chocolate for $0.90. How much does the store charge per ounc
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9514 1404 393
Answer:
- real: -1, 2; complex: +i, -i
- 1, 3, 4
Step-by-step explanation:
1. The graph (red) shows the only real zeros to be -1 and 2. When the corresponding factors are divided from the function, the remaining factor is the quadratic (x^2 +1), which has only complex roots. The quadratic is graphed in green.
The linear factorization is ...
f(x) = (x +1)(x -2)(x -i)(x +i)
The roots are -1, 2, -i, +i.
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2. The graph (blue) shows the zeros are 1, 3, 4.
You observe that the sum of coefficients is zero, so x=1 is a root. Factoring that out gives the quadratic (x^2 -7x +12), which you recognize factors as
(x -3)(x -4) . . . zeros of 3 and 4
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I have attached a spreadsheet that does synthetic division. There are web sites that will do this, too. The tables shown correspond to f1(x)/(x-2) and f2(x)/(x-1). When you fill in the zero and coefficients, the built-in formulas do the rest.
The attached figure represents the cardboard (10inches by 12 inches) and the squares that should be cut to make the box.
let the length of the square = x
So , the length of the box = L = 12 - 2x
the width of the box = W = 10 - 2x
And, the area = L * W = 80 ⇒(given)
∴ L * W = (12-2x)(10-2x) = 80
∴ (12 - 2x)(10-2x) =80
4x² - 44x + 120 = 80 ⇒ multiply the brackets
4x² - 44x +120 - 80 = 0 ⇒ make all variables in one side
4x² - 44x + 40 = 0 ⇒ sum the similar
x² - 11x +10 = 0 ⇒ solve by analysis
(x-1)(x-10) = 0
∴ x = 10 (rejected because the cardboard length = 10 inch)
OR x = 1
∴ the size of the square should be cut from each corner = 1 inch by 1 inch
let the length of the square = x
So , the length of the box = L = 12 - 2x
the width of the box = W = 10 - 2x
And, the area = L * W = 80 ⇒(given)
∴ L * W = (12-2x)(10-2x) = 80
∴ (12 - 2x)(10-2x) =80
4x² - 44x + 120 = 80 ⇒ multiply the brackets
4x² - 44x +120 - 80 = 0 ⇒ make all variables in one side
4x² - 44x + 40 = 0 ⇒ sum the similar
x² - 11x +10 = 0 ⇒ solve by analysis
(x-1)(x-10) = 0
∴ x = 10 (rejected because the cardboard length = 10 inch)
OR x = 1
∴ the size of the square should be cut from each corner = 1 inch by 1 inch