Solve for x: x/2x+3 + 2x+3/x = 184/65

x/2x+3 + 2x +3/x = 184 /65

x(2x + 3)–1 + (2x+3)x-1 = 184/65

(2x + 3)x / (2x +3)x = 184/65

2x² + 3x / 2x² + 3x = 184/65

1x= 184/64

x = 184/64

Step-by-step explanation:

Given that: {x/(2x+3)} + {(2x+3)/x} = 184/65

⇛[{(x*x) + (2x+3)(2x+3)}/{(x)(2x+3)}] = 184/65

⇛[{x² + 2x(2x+3) + 3(2x+3)}/(2x²+3x)] = 184/65

⇛[{x² + 4x² + 6x + 6x + 9}/(2x²+3x)] = 184/65

⇛[{x² + 4x² + 12x + 9}/(2x² + 3x)] = 184/65

⇛{(5x² + 12x + 9)/(2x² + 3x)} = 184/65

On applying cross multiplication then

⇛184(2x² + 3x) = 65(5x² + 12x + 9)

Multiply the number outside of the brackets with numbers and variables in the brackets on LHS and RHS.

⇛388x² + 582x = 325x² + 780x + 584

⇛388x² + 582x -325x² - 780x - 584 = 0

⇛388x²-325x² + 582x-780x - 584 = 0

⇛63x² - 198x - 584 = 0

⇛8(7x² - 22x - 65) = 0

⇛7x² - 22x - 65 = 0

Now,

This is of the form ax² + bx + c = 0, Where, a = 7, b = -22 and c = -65

Using the quadratic formula x = [{-b±√(b²-4ac)}/2a] , we get

x = [{-(-22)±√(-22)² - 4(7)(-65)}/{2(7)]

x = [{-(-22)±√(-22*-22) - 4(7)(-65)}/{2(7)]

x = [{22 ± √(484 + 1820)}/14]

x = [{22 ± √(2304)}/14]

x = {(22 ± 48)/14}

x = {(22 + 48)/14} or {(22 - 48)/14}

x = (70/14) or (-26/14)

x = 5 or x = -13/7

Therefore, x = 5 or -13/7

Answer: Hence, the value of x for the given equation is 5 or -13/7.

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