S4. Mathematics(quadratic&log)

x² + 2x - 5 = 0
α + β = - 2
α β = - 5
6ai)(α + β) (α - β)
= (α + β) (±√(α - β)²)
= (α + β) (±√( (α + β)² - 4 α β ))
= - 2 (±√( ( - 2 )² - 4 (- 5) ))
= - 2 (± 2√6)
= ± 4√6
aii)3^(α+1) ．3^(β+1)
= 3 ^ (α+β+2)
= 3 ^ (-2+2)
= 3 ^ 0
= 1
b)The required equation isx² - (α/β + β/α)x + (α/β)(β/α) = 0
x² - [(α²+β²)/(α β)]x + 1 = 0
x² - [((α+β)² - 2αβ)/(α β)]x + 1 = 0
x² - [((- 2)² - 2(- 5))/(- 5)]x + 1 = 0
x² + (14/5)x + 1 = 0
5x² + 14x + 5 = 0

8a)f(2x + k) - f(2x - k) = x
[(2x + k)² + k] - [(2x - k)² + k] = x
(2x + k)² - (2x - k)² = x
(2x + k - (2x - k)) (2x + k + 2x - k) = x
(2k) (4x) = x
8kx = x
k = 1/8
b)f(2x + k) - k = f(2x - k) + k
f(2x + k) - f(2x - k) = 2k
f(2x + 1/8) - f(2x - 1/8) = 2(1/8)
f(2x + 1/8) - f(2x - 1/8) = 1/4
By the result of a) , f(2x + 1/8) - f(2x - 1/8) = x
So x = 1/4

13)log 18 + log 12 = a + b
log (18*12) = a + b
log 216 = a + b
log 6³ = a + b
3 log 6 = a + b
log 6 = (a + b) / 3

16)log16 4(x+1) - log4 8 = log16 (x - 2)
[log 4(x+1)] / log16 - (log 8) / log 4 = ( log (x - 2) ) / log 16
log 4(x+1) / (2log4) - (log 8) / log 4 = ( log (x - 2) ) / (2log 4)
log 4(x+1) - 2log 8 = log (x - 2)
log [4(x+1)/(x - 2)] = 2log 8
log [4(x+1)/(x - 2)] = log 8²
4(x+1)/(x - 2) = 64
x + 1 = 16(x - 2)
15x - 33 = 0
x = 11/5