Solve the equation using the quadratic formula $2=-\displaystyle\frac { 10 }{ x } -\displaystyle\frac { 5 }{ { x }^{ 2 } } $
- $\left\{ \displaystyle\frac { -5\pm \sqrt { 15 } }{ 2 } \right\} $
- $\left\{ \displaystyle\frac { -5\pm \sqrt { 15 } }{ 4 } \right\} $
- $\left\{ \displaystyle\frac { -5\pm \sqrt { 35 } }{ 2 } \right\} $
- $\left\{ \displaystyle\frac { -10\pm \sqrt { 15 } }{ 2 } \right\} $
Multiply by x^2: 2x^2 = -10x - 5, which is 2x^2 + 10x + 5 = 0. Using the quadratic formula x = (-b +/- sqrt(b^2 - 4ac)) / 2a: x = (-10 +/- sqrt(100 - 40)) / 4 = (-10 +/- sqrt(60)) / 4 = (-10 +/- 2*sqrt(15)) / 4 = (-5 +/- sqrt(15)) / 2.
Multiply the entire equation by x^2 to clear the fractions, resulting in 2x^2 = -10x - 5, and rearrange it into standard form as 2x^2 + 10x + 5 = 0. Applying the quadratic formula, we get x = (-10 plus or minus the square root of (100 - 40)) divided by 4. This simplifies to x = (-10 plus or minus the square root of 60) divided by 4, which reduces to (-5 plus or minus the square root of 15) divided by 2. The solution set is {(-5 plus or minus square root of 15) / 2}.