Solve for x 4/x+1/(x^2)=1/(5x^2)

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

Since ${\displaystyle x,x^2,5x^2}$ contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part ${\displaystyle 1,1,5}$ then find LCM for the variable part ${\displaystyle x^1,x^2,x^2}$.

The LCM is the smallest positive number that all of the numbers divide into evenly.

The number ${\displaystyle 1}$ is not a prime number because it only has one positive factor, which is itself.

Since ${\displaystyle 5}$ has no factors besides ${\displaystyle 1}$ and ${\displaystyle 5}$.

The LCM of ${\displaystyle 1,1,5}$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

The factor for ${\displaystyle x^1}$ is ${\displaystyle x}$ itself.

The factors for ${\displaystyle x^2}$ are ${\displaystyle x\cdot x}$, which is ${\displaystyle x}$ multiplied by each other ${\displaystyle 2}$ times.

The LCM of ${\displaystyle x^1,x^2,x^2}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

Multiply ${\displaystyle x}$ by ${\displaystyle x}$.

The LCM for ${\displaystyle x,x^2,5x^2}$ is the numeric part ${\displaystyle 5}$ multiplied by the variable part.

Multiply each term in ${\displaystyle \frac{4}{x}+\frac{1}{x^2}=\frac{1}{5x^2}}$ by ${\displaystyle 5x^2}$.

Simplify the left side.

Simplify the right side.

Move all terms not containing ${\displaystyle x}$ to the right side of the equation.

Divide each term in ${\displaystyle 20x=-4}$ by ${\displaystyle 20}$ and simplify.