Question

Find all complex solutions for the following equation by hand: 1+3/x=6/x^2

Answer 1

x=-4.37 or 1.37

Explanation:

The equation whose solution is to be found is;

`1 + (3)/(x)= \frac{6}{ {x}^(2) }`

we now multiply through by

`{x}^(2)`

This implies that;

`( {x}^(2))1 +({x}^(2) * (3)/(x))= {x}^(2) * \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 = ( -3\pm √(9+24))/(2)`

`\implies x = (-3-√(33) )/(2)or (-3+√(33))/(2)`

x=-4.37 or 1.37