High School

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5/X + 3 - 3/X - 3= 8/X^2 - 9

Answer :

The solutions to the equation [tex]\frac{5}{x} + 3 - \frac{3}{x} - 3 = \frac{8}{x^2} - 9[/tex] are [tex]x = \frac{-1 + \sqrt{73}}{9}[/tex] and [tex]x = \frac{-1 - \sqrt{73}}{9}[/tex].

To solve the equation [tex]\frac{5}{x} + 3 - \frac{3}{x} - 3 = \frac{8}{x^2} - 9[/tex], we will follow these steps:

  1. Simplify the equation: Observe the left-hand side of the equation:
    [tex]\frac{5}{x} - \frac{3}{x} + 3 - 3 = \frac{2}{x}[/tex]
    Thus, we can rewrite the equation as:
    [tex]\frac{2}{x} = \frac{8}{x^2} - 9[/tex]

  2. Eliminate the fractions: To make it easier to solve, multiply through by [tex]x^2[/tex] (assuming [tex]x \neq 0[/tex]) to eliminate the denominators:
    [tex]2x = 8 - 9x^2[/tex]

  3. Rearranging: Rearranging gives:
    [tex]9x^2 + 2x - 8 = 0[/tex]

  4. Apply the quadratic formula: The quadratic formula can be used here:
    [tex]x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}[/tex]
    Where [tex]a = 9, b = 2, c = -8[/tex].

    • Calculate the discriminant:
      [tex]b^2 - 4ac = 2^2 - 4 \cdot 9 \cdot (-8) = 4 + 288 = 292[/tex]
    • Now apply the formula:
      [tex]x = \frac{-2 \pm \sqrt{292}}{2 \cdot 9}[/tex]
      [tex]x = \frac{-2 \pm 2\sqrt{73}}{18}[/tex]
      [tex]x = \frac{-1 \pm \sqrt{73}}{9}[/tex]

Final solutions:

The solutions are: [tex]x_1 = \frac{-1 + \sqrt{73}}{9}[/tex] and [tex]x_2 = \frac{-1 - \sqrt{73}}{9}[/tex].

Complete question here :-

Solve the equation:

5/X + 3 - 3/X - 3= 8/X^2 - 9

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Rewritten by : Barada