Question

Help its addition and subtraction of Algebraic fractions of different
denominator

Answer 1

`49. \ (x^2)/(x^2 +2 \cdot x - 8) - (x - 4)/(x + 4)`

The above reaction can be rewritten as follows;

`(x^2)/(x^2 +2 \cdot x - 8) - (x - 4)/(x + 4) =(x^2)/((x + 4) \cdot (x - 2)) - (x - 4)/(x + 4) = (x^2 + (x - 2) \cdot (x - 4))/((x + 4) \cdot (x - 2))`

Which gives;

`(x^2)/(x^2 +2 \cdot x - 8) - (x - 4)/(x + 4) = (x^2 -(x^2 -6 \cdot x + 8) )/((x + 4) \cdot (x - 2)) = (6 \cdot x - 8 )/((x + 4) \cdot (x - 2))`

`50. \ (x - 3)/(x^2 +10 \cdot x + 25) + (x - 3)/(x + 5)`

`(x - 3)/(x^2 +10 \cdot x + 25) + (x - 3)/(x + 5) = (x - 3)/((x + 5) \cdot (x + 5)) + (x - 3)/(x + 5) = (x - 3 + (x - 3) \cdot (x + 5))/((x + 5) \cdot (x + 5))`

`(x - 3 + (x - 3) \cdot (x + 5))/((x + 5) \cdot (x + 5)) = (x - 3 + x^2 + 2\cdot x - 15)/((x + 5) \cdot (x + 5)) = ( x^2 + 3 \cdot x - 18)/((x + 5) \cdot (x + 5))`

`53. \ (5)/(a^2 +9 \cdot a + 8) - (3)/(a^2 -6 \cdot a - 16)`

`(5)/(a^2 +9 \cdot a + 8) - (3)/(a^2 -6 \cdot a - 16) = (5)/((a + 1) \cdot (a + 8)) - (3)/((a - 8) \cdot (a + 2) )`

`(5)/((a + 1) \cdot (a + 8)) - (3)/((a - 8) \cdot (a + 2) ) = (5 \cdot (a - 8) \cdot (a + 2) - 3\cdot (a + 1) \cdot (a + 8))/((a + 1) \cdot (a + 8) \cdot (a - 8) \cdot (a + 2))`

`(5 \cdot (a - 8) \cdot (a + 2) - 3\cdot (a + 1) \cdot (a + 8))/((a + 1) \cdot (a + 8) \cdot (a - 8) \cdot (a + 2)) = (2 \cdot a^2 -57 \cdot a -104)/(a^4+3 \cdot a^3-62 \cdot a^2 -192 \cdot a - 1)`

`(5)/(a^2 +9 \cdot a + 8) - (3)/(a^2 -6 \cdot a - 16) = (2 \cdot a^2 -57 \cdot a -104)/(a^4+3 \cdot a^3-62 \cdot a^2 -192 \cdot a - 1)`

`55. \ (2)/(x^2 +6 \cdot x + 9) + (3)/(x^2 + x - 6)`

`(2)/(x^2 +6 \cdot x + 9) + (3)/(x^2 + x - 6) = (2)/((x + 3) \cdot (x + 3)) + (3)/((x+3) \cdot(x - 2))`

`(2)/((x + 3) \cdot (x + 3)) + (3)/((x+3) \cdot(x - 2)) = (2 \cdot(x - 2) + 3\cdot (x + 3) )/((x + 3) \cdot (x + 3) \cdot(x - 2))`

`(2 \cdot(x - 2) + 3\cdot (x + 3) )/((x + 3) \cdot (x + 3) \cdot(x - 2)) = (2 \cdot x - 4 + 3\cdot x + 9 )/((x + 3) \cdot (x + 3) \cdot(x - 2)) = (5 \cdot x + 5 )/((x + 3) \cdot (x + 3) \cdot(x - 2))`
`(5 \cdot x + 5 )/((x + 3) \cdot (x + 3) \cdot(x - 2)) = (5 \cdot x + 5 )/(x ^3 + 4 \cdot x^2-3 \cdot x - 18)`

`57. \ (x)/(2 \cdot x^2 +7 \cdot x + 3) - (3)/(3 \cdot x^2 + 7 \cdot x - 6)`

`(x)/(2 \cdot x^2 +7 \cdot x + 3) - (3)/(3 \cdot x^2 + 7 \cdot x - 6) =(x)/((2 \cdot x + 1) \cdot (x + 3)) - (3)/((3\cdot x-2) \cdot (x + 3))`

`(x)/((2 \cdot x + 1) \cdot (x + 3)) - (3)/((3\cdot x-2) \cdot (x + 3)) = (x \cdot (3 \cdot x - 2) - 3 \cdot (2 \cdot x + 1))/((2 \cdot x + 1) \cdot (x + 3) \cdot (3\cdot x-2))`

`(x \cdot (3 \cdot x - 2) - 3 \cdot (2 \cdot x + 1))/((2 \cdot x + 1) \cdot (x + 3) \cdot (3\cdot x-2)) = ( 3 \cdot x^2 - 8\cdot x - 3 )/((2 \cdot x + 1) \cdot (x + 3) \cdot (3\cdot x-2))`

`( 3 \cdot x^2 - 8\cdot x - 3 )/((2 \cdot x + 1) \cdot (x + 3) \cdot (3\cdot x-2)) = ( (x -3) \cdot (3 \cdot x + 1) )/((2 \cdot x + 1) \cdot (x + 3) \cdot (3\cdot x-2))`

`( (x -3) \cdot (3 \cdot x + 1) )/((2 \cdot x + 1) \cdot (x + 3) \cdot (3\cdot x-2)) = (3 \cdot x^2 - 8 \cdot x -3 )/(6 \cdot x^3+ 17 \cdot x^2 + 5 \cdot x-6)`

`59. \ (x)/(4 \cdot x^2 +11 \cdot x + 6) - (2)/(8 \cdot x^2 + 2 \cdot x - 3)`

Using a graphing calculator, we have;

`(x)/(4 \cdot x^2 +11 \cdot x + 6) - (2)/(8 \cdot x^2 + 2 \cdot x - 3) = (2 \cdot x^2 - 3 \cdot x - 4)/(8 \cdot x^3+18 \cdot x^2+x - 6)`

`61. \ (3 \cdot w+ 12)/(w^2 + w -12) - (2)/(w - 3)`

`(3 \cdot w+ 12)/(w^2 + w -12) - (2)/(w - 3) = (3 \cdot (w+ 4))/((w + 4) \cdot (w - 3)) - (2)/(w - 3) = (3 )/( (w - 3)) - (2)/(w - 3)`

`(3 )/( (w - 3)) - (2)/(w - 3) = (1 )/( (w - 3))`

`61. \ (3 \cdot r)/(2 \cdot r^2 + 10 \cdot r +12) + (3)/(r - 2)`

With the aid of a graphing calculator, we have;

`(3 \cdot r)/(2 \cdot r^2 + 10 \cdot r +12) + (3)/(r - 2) = (3 \cdot r)/(2 \cdot (r+2) \cdot (r + 3)) + (3)/(r - 2)`

`(3 \cdot r)/(2 \cdot (r+2) \cdot (r + 3)) + (3)/(r - 2) = (3 \cdot r \cdot (r - 2) + 3 \cdot 2 \cdot (r+2) \cdot (r + 3))/(2 \cdot (r+2) \cdot (r + 3)\cdot (r - 2) )`

`(3 \cdot r \cdot (r - 2) + 3 \cdot 2 \cdot (r+2) \cdot (r + 3))/(2 \cdot (r+2) \cdot (r + 3)\cdot (r - 2) ) = (9 \cdot r^2 + 24 \cdot r + 36)/(2 \cdot r^3+6\cdot r^2 - 8 \cdot r - 24)`

Explanation: