Answer:
15. Answer: length = 15 ft; width = 3 ft
16. Answer: base = 14 cm; height = 5 cm
17. Answer: x² - 5x - 24 = 0
Step-by-step explanation:
15.
area of rectangle = length * width
length = x - 2
width = x - 14
area = (x - 2) (x - 14)
We are shown the area to be 45 ft².
We set the expression of the area equal to the given area and solve the quadratic equation for x.
(x - 2)(x - 14) = 45
x² - 14x - 2x + 28 - 45 = 0
x² - 16x - 17 = 0
(x - 17)(x + 1) = 0
x - 17 = 0 or x + 1 = 0
x = 17 or x = -1
We discard the negative answer, x = -1, since it would give negative length and width.
x = 17
length = x - 2 = 17 - 2 = 15
width = x - 14 = 17 - 14 = 3
Answer: length = 15 ft; width = 3 ft
16.
area = ½bh
area = ½(x - 5)(x - 14)
area = 35
½(x - 5)(x - 14) = 35
(x - 5)(x - 14) = 70
x² - 14x - 5x + 70 = 70
x² - 19x = 0
x(x - 19) = 0
x = 0 or x - 19 = 0
We discard x = 0 since it would give negative dimensions.
x = 19
base = x - 5 = 19 - 5 = 14
height = x - 14 = 19 - 14 = 5
base = 14
height = 5
Answer: base = 14 cm; height = 5 cm
17.
A polynomial with solutions x = a and x = b can be found by multiplying (x - a) by (x - b). Here, the solutions are x = -3 and x = 8.
[x - (-3)](x - 8) = (x + 3)(x - 8) = x² - 8x + 3x - 24 = x² - 5x - 24
Answer: x² - 5x - 24 = 0
