Find all complex solutions for the following equation

Ответ:

carlosiscr7

20.01.2022

x=-4.37 or 1.37

Step-by-step explanation:

The equation whose solution is to be found is;

$1 + \frac{3}{x}= \frac{6}{ {x}^{2} }$

we now multiply through by

${x}^{2}$

This implies that;

$( {x}^{2})1 +({x}^{2} \times \frac{3}{x})= {x}^{2} \times \frac{6}{ {x}^{2} }$

$\implies{x}^{2}+3x=6$

Subtracting - 6 from both side we obtain;

${x}^{2} + 3x - 6 = 6 - 6$

$\implies{x}^{2} + 3x - 6 =0$ We now obtain a quadratic equation.So, let us use quadratic formula to find the solutions.

Comparing the equation ,

${x}^{2} + 3x - 6 = 0$

to the general quadratic formula,

$a {x}^{2} + bx + c = 0$

we can say that,

a=1,b=3 and c=-6

putting this in to the quadratic formula,

$\implies x = \frac{ -b\pm \sqrt{{b}^{2} - 4ac} }{2a}$ , we obtain

$x = \frac{-3\pm \sqrt{ {3}^{2} - 4(1)(-6)} }{2(1)}$

Simplifying we obtain,

$x = \frac{ -3\pm \sqrt{9+24}}{2}$

$\implies x = \frac{-3-\sqrt{33} }{2}or \frac{-3+\sqrt{33}}{2}$

x=-4.37 or 1.37

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