急 ! 急! F4 Quadratic Equations

4.
The three consecutive positive integers are m, m + 1, m + 2.

(m + 1)(m + 2) = m + 17
m² + 3m + 2 = m + 17
m² + 2m - 15 = 0
(m + 5)(m - 3) = 0
m = -5 (rejected) or m = 3

Hence, the three positive integers are 3, 4 and 5.


=====
5.
Let m and m + 1 be the two consecutive integers.

[1/m] + [1/(m + 1)] = 7/12
[(m + 1)/m(m + 1)] + [m/m(m + 1)] = 7/12
(2m + 1)/m(m + 1) = 7/12
12(2m + 1) = 7m(m + 1)
24m + 12 = 7m² + 7m
7m² - 17m - 12 = 0
(7m + 4)(m - 3) = 0
m = -4/7 (rejected) or m = 3
m + 1 = 4

Hence, the two consecutive integers are 3 and 4.


=====
6.
Let a m x b m be the dimensions of the rectangular lot.

ab = 1200 ...... (1)
b + a + b = 100 ...... (2)

(2) :
a = 100 - 2b ...... (3)

Put (3) into (1) :
(100 - 2b)b = 1200
100b - 2b² = 1200
b² - 50b + 600 = 0
(b - 30)(b - 20) = 0
b = 30 or b = 20

Put b = 30 into (3):
a = 100 - 2*30
a = 40

Put b = 20 into (3):
a = 100 - 2*20
a = 60

Hence, the dimensions are 30m x 40 m or 20 m x 60 m.


=====
7.
Let a and b be the two numbers.

a + b = 19 ...... (1)
a² + b² = 193 ...... (2)

From (1):
a = 19 - b ...... (3)

Put (1) into (3):
(19 - b)² + b² = 193
361 - 38b + b² + b² = 193
2b² - 38b + 168 = 0
b² - 19b + 84 = 0
b = 7 or b = 12

Put b = 7 into (3):
a = 19 - 7
a = 12

Put b = 12 into (3):
a = 19 - 12
a = 7

Hence, the two numbers are 7 and 12.


=====
8.
Let t and u be the tens digit and units digit respectively.

t = u + 1 ...... (1)
(10t + u) - (t² + u²) = 4 ...... (2)

Put (1) into (2):
[10(u + 1) + u] - [(u + 1)² + u²] = 4
10u + 10 + u - u² - 2u - 1 - u² = 4
2u² - 9u + 5 = 0
(2u + 1)(u - 5) = 0
u = -1/2 (rejected) or u = 5

Put u = 5 into (1):
t = 5 + 1
t = 6

Hence, the number is 65.