Respuesta :
Answer:
The following is the expression in simplified form;
[tex]\dfrac{x}{x + 4} -\dfrac{5}{x} = \dfrac{x^2 - 5 \cdot x - 20}{x^2 + 4 \cdot x}[/tex]
The solutions are;
x = 5/2 + √(105) or 5/2 - √(105)
Explanation:
The given expression is written as follows;
[tex]\dfrac{x}{x + 4} -\dfrac{5}{x}[/tex]
We multiply the denominators to form a common denominator and then multiply each individual fraction numerator by the factor of the common denominator from the other fraction as follows;
[tex]\dfrac{x}{x + 4} -\dfrac{5}{x} = \dfrac{x \times x - 5 \times (x + 4)}{(x + 4) \times x}[/tex]
The above expression is simplified to get;
[tex]\dfrac{x^2 - 5 \cdot x - 20}{x^2 + 4 \cdot x}[/tex]
To find the solution of the expression, we equate then expression to zero as follows;
[tex]\dfrac{x^2 - 5 \cdot x + 20}{x^2 + 4 \cdot x} = 0[/tex]
Which gives;
x² - 5·x - 20 = 0 × (x² + 4·x) = 0
x² - 5·x - 20 = 0
Solving with the quadratic equation, [tex]x = \dfrac{-b\pm \sqrt{b^{2}-4\cdot a\cdot c}}{2\cdot a}[/tex] where;
a = 1
b = -5
c = -20
We get;
[tex]x = \dfrac{5\pm \sqrt{(-5)^{2}-4\times 1 \times (-20)}}{2\times 1} = \dfrac{5\pm \sqrt{105}}{2}[/tex]
Therefore, the solutions are;
x = 5/2 + √(105) or 5/2 - √(105).
