2x + 2/x = 3
First we have to find the value of x before looking for what is asked..
(2x)/1 + 2/x = 3/1
Multiply through with the LCM which is x in this case
x((2x)/1) + cancelx(2/cancelx) = x(3/1)
x(2x) + 2 = x(3)
2x^2 + 2 = 3x
2x^2 - 3x + 2 = 0 -> "Quadratic Equation"
Well in solving a Quadratic Equation you have to find the sum and products of the roots..
In this case the possible roots are 4 and 1
In the above equation, you multiply 2 attached to the square of x with the constant..
Hence 2 xx 2 = 4
Therefore, we are going to look for a possible value that can multiply each other to get +4 and can either subtract or add each other to get -3
Hence 4 and 1 are the roots..
Given that rArr 4 xx 1 = 4 and -4 + 1 = -3
Now Solving...
2x^2 color(red)(- 4)x color(red)(+1) x + 1 = 0
(2x^2 - 4x) (- x + 2) = 0
Factorizing..
2x(x - 2) -1(x - 2) = 0
(x - 2) (2x - 1) = 0
x - 2 = 0
x = 2
or
2x - 1 = 0
2x = 1
x= 1/2
Hence we should use x = 2 since it's positive
Now..
Find, x^3 + 1/x + 2
Substitute the value of x into the equation..
x^3 + 1/x + 2 -> "Equation"
color(red)2^3 + 1/color(red)2 + 2
8 + 1/2 +2
8 + 2 + 1/2
10 + 1/2
10 1/2
