Question

Help its addition and subtraction of Algebraic fractions of different denominator

Answer 1

Answer:

$49. \ \dfrac{x^2}{x^2 +2 \cdot x - 8} - \dfrac{x - 4}{x + 4}$

The above reaction can be rewritten as follows;

$\dfrac{x^2}{x^2 +2 \cdot x - 8} - \dfrac{x - 4}{x + 4} =\dfrac{x^2}{(x + 4) \cdot (x - 2)} - \dfrac{x - 4}{x + 4} = \dfrac{x^2 + (x - 2) \cdot (x - 4)}{(x + 4) \cdot (x - 2)}$

Which gives;

$\dfrac{x^2}{x^2 +2 \cdot x - 8} - \dfrac{x - 4}{x + 4} = \dfrac{x^2 -(x^2 -6 \cdot x + 8) }{(x + 4) \cdot (x - 2)} = \dfrac{6 \cdot x - 8 }{(x + 4) \cdot (x - 2)}$

$50. \ \dfrac{x - 3}{x^2 +10 \cdot x + 25} + \dfrac{x - 3}{x + 5}$

$\dfrac{x - 3}{x^2 +10 \cdot x + 25} + \dfrac{x - 3}{x + 5} = \dfrac{x - 3}{(x + 5) \cdot (x + 5)} + \dfrac{x - 3}{x + 5} = \dfrac{x - 3 + (x - 3) \cdot (x + 5)}{(x + 5) \cdot (x + 5)}$

$\dfrac{x - 3 + (x - 3) \cdot (x + 5)}{(x + 5) \cdot (x + 5)} = \dfrac{x - 3 + x^2 + 2\cdot x - 15}{(x + 5) \cdot (x + 5)} = \dfrac{ x^2 + 3 \cdot x - 18}{(x + 5) \cdot (x + 5)}$

$53. \ \dfrac{5}{a^2 +9 \cdot a + 8} - \dfrac{3}{a^2 -6 \cdot a - 16}$

$\dfrac{5}{a^2 +9 \cdot a + 8} - \dfrac{3}{a^2 -6 \cdot a - 16} = \dfrac{5}{(a + 1) \cdot (a + 8)} - \dfrac{3}{(a - 8) \cdot (a + 2) }$

$\dfrac{5}{(a + 1) \cdot (a + 8)} - \dfrac{3}{(a - 8) \cdot (a + 2) } = \dfrac{5 \cdot (a - 8) \cdot (a + 2) - 3\cdot (a + 1) \cdot (a + 8)}{(a + 1) \cdot (a + 8) \cdot (a - 8) \cdot (a + 2)}$

$\dfrac{5 \cdot (a - 8) \cdot (a + 2) - 3\cdot (a + 1) \cdot (a + 8)}{(a + 1) \cdot (a + 8) \cdot (a - 8) \cdot (a + 2)} = \dfrac{2 \cdot a^2 -57 \cdot a -104}{a^4+3 \cdot a^3-62 \cdot a^2 -192 \cdot a - 1}$

$\dfrac{5}{a^2 +9 \cdot a + 8} - \dfrac{3}{a^2 -6 \cdot a - 16} = \dfrac{2 \cdot a^2 -57 \cdot a -104}{a^4+3 \cdot a^3-62 \cdot a^2 -192 \cdot a - 1}$

$55. \ \dfrac{2}{x^2 +6 \cdot x + 9} + \dfrac{3}{x^2 + x - 6}$

$\dfrac{2}{x^2 +6 \cdot x + 9} + \dfrac{3}{x^2 + x - 6} = \dfrac{2}{(x + 3) \cdot (x + 3)} + \dfrac{3}{(x+3) \cdot(x - 2)}$

$\dfrac{2}{(x + 3) \cdot (x + 3)} + \dfrac{3}{(x+3) \cdot(x - 2)} = \dfrac{2 \cdot(x - 2) + 3\cdot (x + 3) }{(x + 3) \cdot (x + 3) \cdot(x - 2)}$

$\dfrac{2 \cdot(x - 2) + 3\cdot (x + 3) }{(x + 3) \cdot (x + 3) \cdot(x - 2)} = \dfrac{2 \cdot x - 4 + 3\cdot x + 9 }{(x + 3) \cdot (x + 3) \cdot(x - 2)} = \dfrac{5 \cdot x + 5 }{(x + 3) \cdot (x + 3) \cdot(x - 2)}$$\dfrac{5 \cdot x + 5 }{(x + 3) \cdot (x + 3) \cdot(x - 2)} = \dfrac{5 \cdot x + 5 }{x ^3 + 4 \cdot x^2-3 \cdot x - 18}$

$57. \ \dfrac{x}{2 \cdot x^2 +7 \cdot x + 3} - \dfrac{3}{3 \cdot x^2 + 7 \cdot x - 6}$

$\dfrac{x}{2 \cdot x^2 +7 \cdot x + 3} - \dfrac{3}{3 \cdot x^2 + 7 \cdot x - 6} =\dfrac{x}{(2 \cdot x + 1) \cdot (x + 3)} - \dfrac{3}{(3\cdot x-2) \cdot (x + 3)}$

$\dfrac{x}{(2 \cdot x + 1) \cdot (x + 3)} - \dfrac{3}{(3\cdot x-2) \cdot (x + 3)} = \dfrac{x \cdot (3 \cdot x - 2) - 3 \cdot (2 \cdot x + 1)}{(2 \cdot x + 1) \cdot (x + 3) \cdot (3\cdot x-2)}$

$\dfrac{x \cdot (3 \cdot x - 2) - 3 \cdot (2 \cdot x + 1)}{(2 \cdot x + 1) \cdot (x + 3) \cdot (3\cdot x-2)} = \dfrac{ 3 \cdot x^2 - 8\cdot x - 3 }{(2 \cdot x + 1) \cdot (x + 3) \cdot (3\cdot x-2)}$

$\dfrac{ 3 \cdot x^2 - 8\cdot x - 3 }{(2 \cdot x + 1) \cdot (x + 3) \cdot (3\cdot x-2)} = \dfrac{ (x -3) \cdot (3 \cdot x + 1) }{(2 \cdot x + 1) \cdot (x + 3) \cdot (3\cdot x-2)}$

$\dfrac{ (x -3) \cdot (3 \cdot x + 1) }{(2 \cdot x + 1) \cdot (x + 3) \cdot (3\cdot x-2)} = \dfrac{3 \cdot x^2 - 8 \cdot x -3 }{6 \cdot x^3+ 17 \cdot x^2 + 5 \cdot x-6}$

$59. \ \dfrac{x}{4 \cdot x^2 +11 \cdot x + 6} - \dfrac{2}{8 \cdot x^2 + 2 \cdot x - 3}$

Using a graphing calculator, we have;

$\dfrac{x}{4 \cdot x^2 +11 \cdot x + 6} - \dfrac{2}{8 \cdot x^2 + 2 \cdot x - 3} = \dfrac{2 \cdot x^2 - 3 \cdot x - 4}{8 \cdot x^3+18 \cdot x^2+x - 6}$

$61. \ \dfrac{3 \cdot w+ 12}{w^2 + w -12} - \dfrac{2}{w - 3}$

$\dfrac{3 \cdot w+ 12}{w^2 + w -12} - \dfrac{2}{w - 3} = \dfrac{3 \cdot (w+ 4)}{(w + 4) \cdot (w - 3)} - \dfrac{2}{w - 3} = \dfrac{3 }{ (w - 3)} - \dfrac{2}{w - 3}$

$\dfrac{3 }{ (w - 3)} - \dfrac{2}{w - 3} = \dfrac{1 }{ (w - 3)}$

$61. \ \dfrac{3 \cdot r}{2 \cdot r^2 + 10 \cdot r +12} + \dfrac{3}{r - 2}$

With the aid of a graphing calculator, we have;

$\dfrac{3 \cdot r}{2 \cdot r^2 + 10 \cdot r +12} + \dfrac{3}{r - 2} = \dfrac{3 \cdot r}{2 \cdot (r+2) \cdot (r + 3)} + \dfrac{3}{r - 2}$

$\dfrac{3 \cdot r}{2 \cdot (r+2) \cdot (r + 3)} + \dfrac{3}{r - 2} = \dfrac{3 \cdot r \cdot (r - 2) + 3 \cdot 2 \cdot (r+2) \cdot (r + 3)}{2 \cdot (r+2) \cdot (r + 3)\cdot (r - 2) }$

$\dfrac{3 \cdot r \cdot (r - 2) + 3 \cdot 2 \cdot (r+2) \cdot (r + 3)}{2 \cdot (r+2) \cdot (r + 3)\cdot (r - 2) } = \dfrac{9 \cdot r^2 + 24 \cdot r + 36}{2 \cdot r^3+6\cdot r^2 - 8 \cdot r - 24}$

Step-by-step explanation: