F.5 Quadratic equation

1.
Let u = log x

(log x)² - 2 log x - 8 = 0
u² - 2u - 8 = 0
(u + 2)(u - 4) = 0
u = -2 or u = 4
log x = -2 or log x= 4
log x = -2 log 10 or log x = 4 log 10
log x = log 10^-2 or log x = log 10^4
x = 10^-2 or x = 10^4
x = 0.01or x = 10000


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2.
Let a the larger number, and b be the smaller number.

a - b = 3 ...... [1]
(√b) + 9 = a ...... [2]

Put [2] into [1] :
[(√b) + 9] - b = 3
√b = b - 6
(√b)² = (b - 6)²
b = b² - 12b + 36
b² - 13b + 36 = 0
(b - 4)(b - 9) = 0
b = 4 or b = 9

Put b = 4 into [2] :
(√4) + 9 = a
a = 11 (rejected for a - b ≠ 3)

Put b = 9 into [2] :
(√9) + 9 = a
a = 12

Hence, the two numbers are 9 and 12.


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3.
[(120/n) - 2](n + 2) = 120 - 8
(2/n)(60 - n)(n + 2) = 112
(60 - n)(n + 2) = 56n
-n² + 58n + 120 = 56n
n² - 2n - 120 = 0
(n + 10)(n - 12) = 0
n = -10(rejected) or n =12