Solve the forlowing rational & fuation [ (5)/(x^2)-4-(8)/(x^2)-1=(2)/(x^2)-3 x+2-(20)/(x^2)-3 x+2 ]

To solve the given rational equation, we first need to simplify and find a common denominator for each side of the equation.<br /><br />The equation is:<br />\[<br />\frac{5}{x^{2}-4} - \frac{8}{x^{2}-1} = \frac{2}{x^{2}-3x+2} - \frac{20}{x^{2}-3x+2}<br />\]<br /><br />First, let's factor each quadratic expression in the denominators:<br /><br />1. \(x^2 - 4\) can be factored as \((x-2)(x+2)\).<br />2. \(x^2 - 1\) can be factored as \((x-1)(x+1)\).<br />3. \(x^2 - 3x + 2\) can be factored as \((x-1)(x-2)\).<br /><br />Now, rewrite the equation with these factored forms:<br />\[<br />\frac{5}{(x-2)(x+2)} - \frac{8}{(x-1)(x+1)} = \frac{2}{(x-1)(x-2)} - \frac{20}{(x-1)(x-2)}<br />\]<br /><br />Simplify the right-hand side:<br />\[<br />\frac{2 - 20}{(x-1)(x-2)} = \frac{-18}{(x-1)(x-2)}<br />\]<br /><br />Now the equation becomes:<br />\[<br />\frac{5}{(x-2)(x+2)} - \frac{8}{(x-1)(x+1)} = \frac{-18}{(x-1)(x-2)}<br />\]<br /><br />To solve this equation, we need a common denominator for all terms. The least common denominator (LCD) is \((x-2)(x+2)(x-1)(x+1)\).<br /><br />Rewrite each term with the LCD:<br />- The first term: \(\frac{5}{(x-2)(x+2)}\) becomes \(\frac{5(x-1)(x+1)}{(x-2)(x+2)(x-1)(x+1)}\).<br />- The second term: \(\frac{8}{(x-1)(x+1)}\) becomes \(\frac{8(x-2)(x+2)}{(x-2)(x+2)(x-1)(x+1)}\).<br />- The third term: \(\frac{-18}{(x-1)(x-2)}\) becomes \(\frac{-18(x+2)}{(x-2)(x+2)(x-1)(x+1)}\).<br /><br />Now, combine the fractions:<br />\[<br />\frac{5(x-1)(x+1) - 8(x-2)(x+2)}{(x-2)(x+2)(x-1)(x+1)} = \frac{-18(x+2)}{(x-2)(x+2)(x-1)(x+1)}<br />\]<br /><br />Since the denominators are the same, equate the numerators:<br />\[<br />5(x-1)(x+1) - 8(x-2)(x+2) = -18(x+2)<br />\]<br /><br />Expand and simplify each term:<br />- \(5(x-1)(x+1) = 5(x^2 - 1) = 5x^2 - 5\)<br />- \(8(x-2)(x+2) = 8(x^2 - 4) = 8x^2 - 32\)<br />- \(-18(x+2) = -18x - 36\)<br /><br />Substitute back into the equation:<br />\[<br />5x^2 - 5 - (8x^2 - 32) = -18x - 36<br />\]<br /><br />Simplify:<br />\[<br />5x^2 - 5 - 8x^2 + 32 = -18x - 36<br />\]<br />\[<br />-3x^2 + 27 = -18x - 36<br />\]<br /><br />Rearrange all terms to one side:<br />\[<br />-3x^2 + 18x + 63 = 0<br />\]<br /><br />Divide the entire equation by \(-3\) to simplify:<br />\[<br />x^2 - 6x - 21 = 0<br />\]<br /><br />Solve this quadratic equation using the quadratic formula:<br />\[<br />x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}<br />\]<br />where \(a = 1\), \(b = -6\), and \(c = -21\).<br /><br />Calculate the discriminant:<br />\[<br />b^2 - 4ac = (-6)^2 - 4(1)(-21) = 36 + 84 = 120<br />\]<br /><br />Thus, the solutions are:<br />\[<br />x = \frac{6 \pm \sqrt{120}}{2}<br />\]<br /><br />Simplify \(\sqrt{120}\):<br />\[<br />\sqrt{120} = \sqrt{4 \times 30} = 2\sqrt{30}<br />\]<br /><br />So the solutions are:<br />\[<br />x = \frac{6 \pm 2\sqrt{30}}{2}<br />\]<br />\[<br />x = 3 \pm \sqrt{30}<br />\]<br /><br />Therefore, the solutions to the equation are \(x = 3 + \sqrt{30}\) and \(x = 3 - \sqrt{30}\).