Thursday, January 1, 2015

simplify the following: {x^2 +8x +7}/{x^2 +6x +9} *{ x^2-13x-18}/{x^2 -5x -6}

We'll determine the roots of the numerators and
denominators of both fractions:


x^2 +8x +7 =
0


We notice that the sum of the roots is -8 and the product
is 7, then the roots are: x1 = -7 and x2 = -1


We can write
the quadratic as a product of linear factors:


x^2 +8x +7 =
(x+1)(x+7)


In the case of  denominator of the 1st fraction,
we'll recognize the perfect square:


x^2 +6x +9 =
(x+3)^2


We'll find the roots of the numerator of the 2nd
fraction:


x^2 - 13x - 18 =
0


x1 = [13+sqrt(169+72)]/2


x1
= [13+sqrt(241)]/2


x2 =
[13-sqrt(241)]/2


x^2 - 13x - 18 = (x - 13/2 - sqrt241/2)(x
- 13/2 + sqrt241/2)


We'll determine the roots of the 2nd
denominator:


x^2 - 5x - 6 =
0


x1 = -1, x2 = 6


x^2 - 5x - 6
= (x-6)(x+1)


The product of fractions will
be:


(x+1)*(x+7)*(x - 13/2 - sqrt241/2)*(x - 13/2 +
sqrt241/2)/(x+3)^2*(x-6)*(x+1)


We'll simplify
and we'll get: (x+7)*(x^2 - 13x -
18)/(x+3)^2*(x-6)

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