Quardratic equation

1.
7/(x - 4)(x + 3) - 4/(x + 3)(x - 1)
= 7(x - 1)/(x - 4)(x + 3)(x - 1) - 4(x - 4)/(x - 4)(x + 3)(x - 1)
= (7x - 7 - 4x + 16)/(x - 4)(x + 3)(x - 1)
= 3(x + 3)/(x - 4)(x + 3)(x - 1)
= 3/(x - 4)(x - 1)

7/(x - 4)(x + 3) - 4/(x + 3)(x - 1) = 3/2
3/(x - 4)(x - 1) = 3/2
(x - 4)(x - 1) = 2
x² - 5x + 2 = 0
x = {5 ± √[(-5)² - 4(1)(2)]}/2
x = (5 + √17)/2 or x = (5 - √17)/2


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2.
100/V - 100/(V + 10) = 5/6
[100(V + 10) - 100V] / V(V + 10) = 5/6
5V(V + 10) = 6000
V² + 10V - 1200 = 0
(V - 30)(V + 40) = 0
V = 30 or V = 40 (rejected)


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3.
Let n be the number of toys bought.

[(40/n) + 1](n - 2) = 40
(40 + n)(n - 2) = 40n
40n - 80 + n² - 2n = 40n
n² - 2n - 80 = 0
(n - 10)(n + 8) = 0
n = 10 or n = -8 (rejected)

Number of toys bought = 10


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4.
Let v miles per hour be the normal speed of the inter-city express.

200/(v - 10) - 200/v = 40/60
[200v - 200(v - 10)] / v(v - 10) = 2/3
2v(v - 10) = 3(2000)
v² - 10v - 3000 = 0
(v - 60)(v + 50) = 0
v = 60 or v = -50 (rejected)

The normal speed of the inter-city express = 60 miles/hour


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5.
(m + 1)/(n + 1) =3/2 … (1)
(m² - 1)/(n² - 1) = 21/8 … (2)

(1):
2m + 2 = 3n + 3
2m = 3n + 1
m = (3n + 1)/2 … (3)

(2):
8m² - 8 = 21n² - 21
8m² = 21n² - 13 … (4)

Put (3) into (4):
8[(3n + 1)/2]² = 21n² - 13
18n² + 12n +2 = 21n² - 13
3n² - 12n - 15 = 0
n² - 4n - 5 = 0
(n - 5)(n + 1) = 0
n = 5 or n = -1 (rejected)

Put n = 5 into (3):
m = 8

Hence, m = 8 and n = 5


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6.
(x - 1)/(2x - 3) = (3x + 2)/(x + 1)
(x - 1)(x + 1) = (3x+ 2)(2x - 3)
x² - 1 = 6x² - 5x - 6
5x² - 5x - 5 = 0x² - x - 1 = 0x = {-(-1) ±√[(1)² - 4(1)(-1)]}/2(1)
x = (1 ± √5)/2
x = 1.62 or x = -0.62


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7.
(4x + 3)/(2x - 1) + (6x + 1)/(2x + 1) = 3
[(4x + 3)(2x + 1) + (6x + 1)(2x - 1)] / (2x - 1)(2x + 1) = 3
8x² + 10x + 3 + 12x² - 4x - 1 = 3(4x² - 1)
20x² + 6x + 2 = 12x² - 3
8x² + 4x + 5 = 0

Determinant, Δ
= (4)² - 4(8)(5)
= -144 < 0

Since Δ < 0, the equation has no real roots.